Sr Examen
Integral de 2y/cos^2(y)^2 dx
Solución
Ha introducido
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1 / | | 2*y | ------- dy | 4 | cos (y) | / 0
$$\int\limits_{0}^{1} \frac{2 y}{\cos^{4}{\left(y \right)}}\, dy$$
Integral((2*y)/cos(y)^4, (y, 0, 1))
Respuesta (Indefinida)
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/ 4/y\ / /y\\ / /y\\ 2/y\ / 2/y\\ 5/y\ /y\ 2/y\ / 2/y\\ 4/y\ / /y\\ 4/y\ / /y\\ 6/y\ / 2/y\\ 6/y\ / /y\\ 6/y\ / /y\\ 3/y\ 2/y\ / /y\\ 2/y\ / /y\\ 4/y\ / 2/y\\ | 4*tan |-| 4*log|1 + tan|-|| 4*log|-1 + tan|-|| 4*tan |-| 4*log|1 + tan |-|| 12*y*tan |-| 12*y*tan|-| 12*tan |-|*log|1 + tan |-|| 12*tan |-|*log|1 + tan|-|| 12*tan |-|*log|-1 + tan|-|| 4*tan |-|*log|1 + tan |-|| 4*tan |-|*log|1 + tan|-|| 4*tan |-|*log|-1 + tan|-|| 8*y*tan |-| 12*tan |-|*log|1 + tan|-|| 12*tan |-|*log|-1 + tan|-|| 12*tan |-|*log|1 + tan |-|| | 2*y \2/ \ \2// \ \2// \2/ \ \2// \2/ \2/ \2/ \ \2// \2/ \ \2// \2/ \ \2// \2/ \ \2// \2/ \ \2// \2/ \ \2// \2/ \2/ \ \2// \2/ \ \2// \2/ \ \2// | ------- dy = C - -------------------------------------- - -------------------------------------- - -------------------------------------- + -------------------------------------- + -------------------------------------- - -------------------------------------- - -------------------------------------- - -------------------------------------- - -------------------------------------- - -------------------------------------- - -------------------------------------- + -------------------------------------- + -------------------------------------- + -------------------------------------- + -------------------------------------- + -------------------------------------- + -------------------------------------- | 4 4/y\ 6/y\ 2/y\ 4/y\ 6/y\ 2/y\ 4/y\ 6/y\ 2/y\ 4/y\ 6/y\ 2/y\ 4/y\ 6/y\ 2/y\ 4/y\ 6/y\ 2/y\ 4/y\ 6/y\ 2/y\ 4/y\ 6/y\ 2/y\ 4/y\ 6/y\ 2/y\ 4/y\ 6/y\ 2/y\ 4/y\ 6/y\ 2/y\ 4/y\ 6/y\ 2/y\ 4/y\ 6/y\ 2/y\ 4/y\ 6/y\ 2/y\ 4/y\ 6/y\ 2/y\ 4/y\ 6/y\ 2/y\ 4/y\ 6/y\ 2/y\ | cos (y) -3 - 9*tan |-| + 3*tan |-| + 9*tan |-| -3 - 9*tan |-| + 3*tan |-| + 9*tan |-| -3 - 9*tan |-| + 3*tan |-| + 9*tan |-| -3 - 9*tan |-| + 3*tan |-| + 9*tan |-| -3 - 9*tan |-| + 3*tan |-| + 9*tan |-| -3 - 9*tan |-| + 3*tan |-| + 9*tan |-| -3 - 9*tan |-| + 3*tan |-| + 9*tan |-| -3 - 9*tan |-| + 3*tan |-| + 9*tan |-| -3 - 9*tan |-| + 3*tan |-| + 9*tan |-| -3 - 9*tan |-| + 3*tan |-| + 9*tan |-| -3 - 9*tan |-| + 3*tan |-| + 9*tan |-| -3 - 9*tan |-| + 3*tan |-| + 9*tan |-| -3 - 9*tan |-| + 3*tan |-| + 9*tan |-| -3 - 9*tan |-| + 3*tan |-| + 9*tan |-| -3 - 9*tan |-| + 3*tan |-| + 9*tan |-| -3 - 9*tan |-| + 3*tan |-| + 9*tan |-| -3 - 9*tan |-| + 3*tan |-| + 9*tan |-| | \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ /
$$\int \frac{2 y}{\cos^{4}{\left(y \right)}}\, dy = C - \frac{12 y \tan^{5}{\left(\frac{y}{2} \right)}}{3 \tan^{6}{\left(\frac{y}{2} \right)} - 9 \tan^{4}{\left(\frac{y}{2} \right)} + 9 \tan^{2}{\left(\frac{y}{2} \right)} - 3} + \frac{8 y \tan^{3}{\left(\frac{y}{2} \right)}}{3 \tan^{6}{\left(\frac{y}{2} \right)} - 9 \tan^{4}{\left(\frac{y}{2} \right)} + 9 \tan^{2}{\left(\frac{y}{2} \right)} - 3} - \frac{12 y \tan{\left(\frac{y}{2} \right)}}{3 \tan^{6}{\left(\frac{y}{2} \right)} - 9 \tan^{4}{\left(\frac{y}{2} \right)} + 9 \tan^{2}{\left(\frac{y}{2} \right)} - 3} + \frac{4 \log{\left(\tan{\left(\frac{y}{2} \right)} - 1 \right)} \tan^{6}{\left(\frac{y}{2} \right)}}{3 \tan^{6}{\left(\frac{y}{2} \right)} - 9 \tan^{4}{\left(\frac{y}{2} \right)} + 9 \tan^{2}{\left(\frac{y}{2} \right)} - 3} - \frac{12 \log{\left(\tan{\left(\frac{y}{2} \right)} - 1 \right)} \tan^{4}{\left(\frac{y}{2} \right)}}{3 \tan^{6}{\left(\frac{y}{2} \right)} - 9 \tan^{4}{\left(\frac{y}{2} \right)} + 9 \tan^{2}{\left(\frac{y}{2} \right)} - 3} + \frac{12 \log{\left(\tan{\left(\frac{y}{2} \right)} - 1 \right)} \tan^{2}{\left(\frac{y}{2} \right)}}{3 \tan^{6}{\left(\frac{y}{2} \right)} - 9 \tan^{4}{\left(\frac{y}{2} \right)} + 9 \tan^{2}{\left(\frac{y}{2} \right)} - 3} - \frac{4 \log{\left(\tan{\left(\frac{y}{2} \right)} - 1 \right)}}{3 \tan^{6}{\left(\frac{y}{2} \right)} - 9 \tan^{4}{\left(\frac{y}{2} \right)} + 9 \tan^{2}{\left(\frac{y}{2} \right)} - 3} + \frac{4 \log{\left(\tan{\left(\frac{y}{2} \right)} + 1 \right)} \tan^{6}{\left(\frac{y}{2} \right)}}{3 \tan^{6}{\left(\frac{y}{2} \right)} - 9 \tan^{4}{\left(\frac{y}{2} \right)} + 9 \tan^{2}{\left(\frac{y}{2} \right)} - 3} - \frac{12 \log{\left(\tan{\left(\frac{y}{2} \right)} + 1 \right)} \tan^{4}{\left(\frac{y}{2} \right)}}{3 \tan^{6}{\left(\frac{y}{2} \right)} - 9 \tan^{4}{\left(\frac{y}{2} \right)} + 9 \tan^{2}{\left(\frac{y}{2} \right)} - 3} + \frac{12 \log{\left(\tan{\left(\frac{y}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{y}{2} \right)}}{3 \tan^{6}{\left(\frac{y}{2} \right)} - 9 \tan^{4}{\left(\frac{y}{2} \right)} + 9 \tan^{2}{\left(\frac{y}{2} \right)} - 3} - \frac{4 \log{\left(\tan{\left(\frac{y}{2} \right)} + 1 \right)}}{3 \tan^{6}{\left(\frac{y}{2} \right)} - 9 \tan^{4}{\left(\frac{y}{2} \right)} + 9 \tan^{2}{\left(\frac{y}{2} \right)} - 3} - \frac{4 \log{\left(\tan^{2}{\left(\frac{y}{2} \right)} + 1 \right)} \tan^{6}{\left(\frac{y}{2} \right)}}{3 \tan^{6}{\left(\frac{y}{2} \right)} - 9 \tan^{4}{\left(\frac{y}{2} \right)} + 9 \tan^{2}{\left(\frac{y}{2} \right)} - 3} + \frac{12 \log{\left(\tan^{2}{\left(\frac{y}{2} \right)} + 1 \right)} \tan^{4}{\left(\frac{y}{2} \right)}}{3 \tan^{6}{\left(\frac{y}{2} \right)} - 9 \tan^{4}{\left(\frac{y}{2} \right)} + 9 \tan^{2}{\left(\frac{y}{2} \right)} - 3} - \frac{12 \log{\left(\tan^{2}{\left(\frac{y}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{y}{2} \right)}}{3 \tan^{6}{\left(\frac{y}{2} \right)} - 9 \tan^{4}{\left(\frac{y}{2} \right)} + 9 \tan^{2}{\left(\frac{y}{2} \right)} - 3} + \frac{4 \log{\left(\tan^{2}{\left(\frac{y}{2} \right)} + 1 \right)}}{3 \tan^{6}{\left(\frac{y}{2} \right)} - 9 \tan^{4}{\left(\frac{y}{2} \right)} + 9 \tan^{2}{\left(\frac{y}{2} \right)} - 3} - \frac{4 \tan^{4}{\left(\frac{y}{2} \right)}}{3 \tan^{6}{\left(\frac{y}{2} \right)} - 9 \tan^{4}{\left(\frac{y}{2} \right)} + 9 \tan^{2}{\left(\frac{y}{2} \right)} - 3} + \frac{4 \tan^{2}{\left(\frac{y}{2} \right)}}{3 \tan^{6}{\left(\frac{y}{2} \right)} - 9 \tan^{4}{\left(\frac{y}{2} \right)} + 9 \tan^{2}{\left(\frac{y}{2} \right)} - 3}$$
Gráfica
Respuesta
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5 4 2 / 2 \ 3 2 / 2 \ 4 4 6 / 2 \ 6 6 2 2 4 / 2 \
12*tan (1/2) 12*tan(1/2) 4*tan (1/2) 4*(pi*I + log(1 - tan(1/2))) 4*log(1 + tan(1/2)) 4*tan (1/2) 4*log\1 + tan (1/2)/ 8*tan (1/2) 4*pi*I 12*tan (1/2)*log\1 + tan (1/2)/ 12*tan (1/2)*(pi*I + log(1 - tan(1/2))) 12*tan (1/2)*log(1 + tan(1/2)) 4*tan (1/2)*log\1 + tan (1/2)/ 4*tan (1/2)*(pi*I + log(1 - tan(1/2))) 4*tan (1/2)*log(1 + tan(1/2)) 12*tan (1/2)*(pi*I + log(1 - tan(1/2))) 12*tan (1/2)*log(1 + tan(1/2)) 12*tan (1/2)*log\1 + tan (1/2)/
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4 6 2 4 6 2 4 6 2 4 6 2 4 6 2 4 6 2 4 6 2 4 6 2 3 4 6 2 4 6 2 4 6 2 4 6 2 4 6 2 4 6 2 4 6 2 4 6 2 4 6 2
-3 - 9*tan (1/2) + 3*tan (1/2) + 9*tan (1/2) -3 - 9*tan (1/2) + 3*tan (1/2) + 9*tan (1/2) -3 - 9*tan (1/2) + 3*tan (1/2) + 9*tan (1/2) -3 - 9*tan (1/2) + 3*tan (1/2) + 9*tan (1/2) -3 - 9*tan (1/2) + 3*tan (1/2) + 9*tan (1/2) -3 - 9*tan (1/2) + 3*tan (1/2) + 9*tan (1/2) -3 - 9*tan (1/2) + 3*tan (1/2) + 9*tan (1/2) -3 - 9*tan (1/2) + 3*tan (1/2) + 9*tan (1/2) -3 - 9*tan (1/2) + 3*tan (1/2) + 9*tan (1/2) -3 - 9*tan (1/2) + 3*tan (1/2) + 9*tan (1/2) -3 - 9*tan (1/2) + 3*tan (1/2) + 9*tan (1/2) -3 - 9*tan (1/2) + 3*tan (1/2) + 9*tan (1/2) -3 - 9*tan (1/2) + 3*tan (1/2) + 9*tan (1/2) -3 - 9*tan (1/2) + 3*tan (1/2) + 9*tan (1/2) -3 - 9*tan (1/2) + 3*tan (1/2) + 9*tan (1/2) -3 - 9*tan (1/2) + 3*tan (1/2) + 9*tan (1/2) -3 - 9*tan (1/2) + 3*tan (1/2) + 9*tan (1/2)
$$\frac{12 \log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{-3 - 9 \tan^{4}{\left(\frac{1}{2} \right)} + 3 \tan^{6}{\left(\frac{1}{2} \right)} + 9 \tan^{2}{\left(\frac{1}{2} \right)}} + \frac{8 \tan^{3}{\left(\frac{1}{2} \right)}}{-3 - 9 \tan^{4}{\left(\frac{1}{2} \right)} + 3 \tan^{6}{\left(\frac{1}{2} \right)} + 9 \tan^{2}{\left(\frac{1}{2} \right)}} + \frac{4 \tan^{2}{\left(\frac{1}{2} \right)}}{-3 - 9 \tan^{4}{\left(\frac{1}{2} \right)} + 3 \tan^{6}{\left(\frac{1}{2} \right)} + 9 \tan^{2}{\left(\frac{1}{2} \right)}} + \frac{4 \log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)}}{-3 - 9 \tan^{4}{\left(\frac{1}{2} \right)} + 3 \tan^{6}{\left(\frac{1}{2} \right)} + 9 \tan^{2}{\left(\frac{1}{2} \right)}} + \frac{12 \log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{4}{\left(\frac{1}{2} \right)}}{-3 - 9 \tan^{4}{\left(\frac{1}{2} \right)} + 3 \tan^{6}{\left(\frac{1}{2} \right)} + 9 \tan^{2}{\left(\frac{1}{2} \right)}} + \frac{4 \log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{6}{\left(\frac{1}{2} \right)}}{-3 - 9 \tan^{4}{\left(\frac{1}{2} \right)} + 3 \tan^{6}{\left(\frac{1}{2} \right)} + 9 \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{4 \log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{6}{\left(\frac{1}{2} \right)}}{-3 - 9 \tan^{4}{\left(\frac{1}{2} \right)} + 3 \tan^{6}{\left(\frac{1}{2} \right)} + 9 \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{4 \tan^{4}{\left(\frac{1}{2} \right)}}{-3 - 9 \tan^{4}{\left(\frac{1}{2} \right)} + 3 \tan^{6}{\left(\frac{1}{2} \right)} + 9 \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{12 \log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{4}{\left(\frac{1}{2} \right)}}{-3 - 9 \tan^{4}{\left(\frac{1}{2} \right)} + 3 \tan^{6}{\left(\frac{1}{2} \right)} + 9 \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{12 \tan^{5}{\left(\frac{1}{2} \right)}}{-3 - 9 \tan^{4}{\left(\frac{1}{2} \right)} + 3 \tan^{6}{\left(\frac{1}{2} \right)} + 9 \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{12 \log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{-3 - 9 \tan^{4}{\left(\frac{1}{2} \right)} + 3 \tan^{6}{\left(\frac{1}{2} \right)} + 9 \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{4 \log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)}}{-3 - 9 \tan^{4}{\left(\frac{1}{2} \right)} + 3 \tan^{6}{\left(\frac{1}{2} \right)} + 9 \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{12 \tan{\left(\frac{1}{2} \right)}}{-3 - 9 \tan^{4}{\left(\frac{1}{2} \right)} + 3 \tan^{6}{\left(\frac{1}{2} \right)} + 9 \tan^{2}{\left(\frac{1}{2} \right)}} + \frac{12 \left(\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} + i \pi\right) \tan^{2}{\left(\frac{1}{2} \right)}}{-3 - 9 \tan^{4}{\left(\frac{1}{2} \right)} + 3 \tan^{6}{\left(\frac{1}{2} \right)} + 9 \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{4 i \pi}{3} + \frac{4 \left(\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} + i \pi\right) \tan^{6}{\left(\frac{1}{2} \right)}}{-3 - 9 \tan^{4}{\left(\frac{1}{2} \right)} + 3 \tan^{6}{\left(\frac{1}{2} \right)} + 9 \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{12 \left(\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} + i \pi\right) \tan^{4}{\left(\frac{1}{2} \right)}}{-3 - 9 \tan^{4}{\left(\frac{1}{2} \right)} + 3 \tan^{6}{\left(\frac{1}{2} \right)} + 9 \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{4 \left(\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} + i \pi\right)}{-3 - 9 \tan^{4}{\left(\frac{1}{2} \right)} + 3 \tan^{6}{\left(\frac{1}{2} \right)} + 9 \tan^{2}{\left(\frac{1}{2} \right)}}$$
=
=
5 4 2 / 2 \ 3 2 / 2 \ 4 4 6 / 2 \ 6 6 2 2 4 / 2 \
12*tan (1/2) 12*tan(1/2) 4*tan (1/2) 4*(pi*I + log(1 - tan(1/2))) 4*log(1 + tan(1/2)) 4*tan (1/2) 4*log\1 + tan (1/2)/ 8*tan (1/2) 4*pi*I 12*tan (1/2)*log\1 + tan (1/2)/ 12*tan (1/2)*(pi*I + log(1 - tan(1/2))) 12*tan (1/2)*log(1 + tan(1/2)) 4*tan (1/2)*log\1 + tan (1/2)/ 4*tan (1/2)*(pi*I + log(1 - tan(1/2))) 4*tan (1/2)*log(1 + tan(1/2)) 12*tan (1/2)*(pi*I + log(1 - tan(1/2))) 12*tan (1/2)*log(1 + tan(1/2)) 12*tan (1/2)*log\1 + tan (1/2)/
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4 6 2 4 6 2 4 6 2 4 6 2 4 6 2 4 6 2 4 6 2 4 6 2 3 4 6 2 4 6 2 4 6 2 4 6 2 4 6 2 4 6 2 4 6 2 4 6 2 4 6 2
-3 - 9*tan (1/2) + 3*tan (1/2) + 9*tan (1/2) -3 - 9*tan (1/2) + 3*tan (1/2) + 9*tan (1/2) -3 - 9*tan (1/2) + 3*tan (1/2) + 9*tan (1/2) -3 - 9*tan (1/2) + 3*tan (1/2) + 9*tan (1/2) -3 - 9*tan (1/2) + 3*tan (1/2) + 9*tan (1/2) -3 - 9*tan (1/2) + 3*tan (1/2) + 9*tan (1/2) -3 - 9*tan (1/2) + 3*tan (1/2) + 9*tan (1/2) -3 - 9*tan (1/2) + 3*tan (1/2) + 9*tan (1/2) -3 - 9*tan (1/2) + 3*tan (1/2) + 9*tan (1/2) -3 - 9*tan (1/2) + 3*tan (1/2) + 9*tan (1/2) -3 - 9*tan (1/2) + 3*tan (1/2) + 9*tan (1/2) -3 - 9*tan (1/2) + 3*tan (1/2) + 9*tan (1/2) -3 - 9*tan (1/2) + 3*tan (1/2) + 9*tan (1/2) -3 - 9*tan (1/2) + 3*tan (1/2) + 9*tan (1/2) -3 - 9*tan (1/2) + 3*tan (1/2) + 9*tan (1/2) -3 - 9*tan (1/2) + 3*tan (1/2) + 9*tan (1/2) -3 - 9*tan (1/2) + 3*tan (1/2) + 9*tan (1/2)
$$\frac{12 \log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{-3 - 9 \tan^{4}{\left(\frac{1}{2} \right)} + 3 \tan^{6}{\left(\frac{1}{2} \right)} + 9 \tan^{2}{\left(\frac{1}{2} \right)}} + \frac{8 \tan^{3}{\left(\frac{1}{2} \right)}}{-3 - 9 \tan^{4}{\left(\frac{1}{2} \right)} + 3 \tan^{6}{\left(\frac{1}{2} \right)} + 9 \tan^{2}{\left(\frac{1}{2} \right)}} + \frac{4 \tan^{2}{\left(\frac{1}{2} \right)}}{-3 - 9 \tan^{4}{\left(\frac{1}{2} \right)} + 3 \tan^{6}{\left(\frac{1}{2} \right)} + 9 \tan^{2}{\left(\frac{1}{2} \right)}} + \frac{4 \log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)}}{-3 - 9 \tan^{4}{\left(\frac{1}{2} \right)} + 3 \tan^{6}{\left(\frac{1}{2} \right)} + 9 \tan^{2}{\left(\frac{1}{2} \right)}} + \frac{12 \log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{4}{\left(\frac{1}{2} \right)}}{-3 - 9 \tan^{4}{\left(\frac{1}{2} \right)} + 3 \tan^{6}{\left(\frac{1}{2} \right)} + 9 \tan^{2}{\left(\frac{1}{2} \right)}} + \frac{4 \log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{6}{\left(\frac{1}{2} \right)}}{-3 - 9 \tan^{4}{\left(\frac{1}{2} \right)} + 3 \tan^{6}{\left(\frac{1}{2} \right)} + 9 \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{4 \log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{6}{\left(\frac{1}{2} \right)}}{-3 - 9 \tan^{4}{\left(\frac{1}{2} \right)} + 3 \tan^{6}{\left(\frac{1}{2} \right)} + 9 \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{4 \tan^{4}{\left(\frac{1}{2} \right)}}{-3 - 9 \tan^{4}{\left(\frac{1}{2} \right)} + 3 \tan^{6}{\left(\frac{1}{2} \right)} + 9 \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{12 \log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{4}{\left(\frac{1}{2} \right)}}{-3 - 9 \tan^{4}{\left(\frac{1}{2} \right)} + 3 \tan^{6}{\left(\frac{1}{2} \right)} + 9 \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{12 \tan^{5}{\left(\frac{1}{2} \right)}}{-3 - 9 \tan^{4}{\left(\frac{1}{2} \right)} + 3 \tan^{6}{\left(\frac{1}{2} \right)} + 9 \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{12 \log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{-3 - 9 \tan^{4}{\left(\frac{1}{2} \right)} + 3 \tan^{6}{\left(\frac{1}{2} \right)} + 9 \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{4 \log{\left(\tan{\left(\frac{1}{2} \right)} + 1 \right)}}{-3 - 9 \tan^{4}{\left(\frac{1}{2} \right)} + 3 \tan^{6}{\left(\frac{1}{2} \right)} + 9 \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{12 \tan{\left(\frac{1}{2} \right)}}{-3 - 9 \tan^{4}{\left(\frac{1}{2} \right)} + 3 \tan^{6}{\left(\frac{1}{2} \right)} + 9 \tan^{2}{\left(\frac{1}{2} \right)}} + \frac{12 \left(\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} + i \pi\right) \tan^{2}{\left(\frac{1}{2} \right)}}{-3 - 9 \tan^{4}{\left(\frac{1}{2} \right)} + 3 \tan^{6}{\left(\frac{1}{2} \right)} + 9 \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{4 i \pi}{3} + \frac{4 \left(\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} + i \pi\right) \tan^{6}{\left(\frac{1}{2} \right)}}{-3 - 9 \tan^{4}{\left(\frac{1}{2} \right)} + 3 \tan^{6}{\left(\frac{1}{2} \right)} + 9 \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{12 \left(\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} + i \pi\right) \tan^{4}{\left(\frac{1}{2} \right)}}{-3 - 9 \tan^{4}{\left(\frac{1}{2} \right)} + 3 \tan^{6}{\left(\frac{1}{2} \right)} + 9 \tan^{2}{\left(\frac{1}{2} \right)}} - \frac{4 \left(\log{\left(1 - \tan{\left(\frac{1}{2} \right)} \right)} + i \pi\right)}{-3 - 9 \tan^{4}{\left(\frac{1}{2} \right)} + 3 \tan^{6}{\left(\frac{1}{2} \right)} + 9 \tan^{2}{\left(\frac{1}{2} \right)}}$$
-12*tan(1/2)^5/(-3 - 9*tan(1/2)^4 + 3*tan(1/2)^6 + 9*tan(1/2)^2) - 12*tan(1/2)/(-3 - 9*tan(1/2)^4 + 3*tan(1/2)^6 + 9*tan(1/2)^2) - 4*tan(1/2)^4/(-3 - 9*tan(1/2)^4 + 3*tan(1/2)^6 + 9*tan(1/2)^2) - 4*(pi*i + log(1 - tan(1/2)))/(-3 - 9*tan(1/2)^4 + 3*tan(1/2)^6 + 9*tan(1/2)^2) - 4*log(1 + tan(1/2))/(-3 - 9*tan(1/2)^4 + 3*tan(1/2)^6 + 9*tan(1/2)^2) + 4*tan(1/2)^2/(-3 - 9*tan(1/2)^4 + 3*tan(1/2)^6 + 9*tan(1/2)^2) + 4*log(1 + tan(1/2)^2)/(-3 - 9*tan(1/2)^4 + 3*tan(1/2)^6 + 9*tan(1/2)^2) + 8*tan(1/2)^3/(-3 - 9*tan(1/2)^4 + 3*tan(1/2)^6 + 9*tan(1/2)^2) - 4*pi*i/3 - 12*tan(1/2)^2*log(1 + tan(1/2)^2)/(-3 - 9*tan(1/2)^4 + 3*tan(1/2)^6 + 9*tan(1/2)^2) - 12*tan(1/2)^4*(pi*i + log(1 - tan(1/2)))/(-3 - 9*tan(1/2)^4 + 3*tan(1/2)^6 + 9*tan(1/2)^2) - 12*tan(1/2)^4*log(1 + tan(1/2))/(-3 - 9*tan(1/2)^4 + 3*tan(1/2)^6 + 9*tan(1/2)^2) - 4*tan(1/2)^6*log(1 + tan(1/2)^2)/(-3 - 9*tan(1/2)^4 + 3*tan(1/2)^6 + 9*tan(1/2)^2) + 4*tan(1/2)^6*(pi*i + log(1 - tan(1/2)))/(-3 - 9*tan(1/2)^4 + 3*tan(1/2)^6 + 9*tan(1/2)^2) + 4*tan(1/2)^6*log(1 + tan(1/2))/(-3 - 9*tan(1/2)^4 + 3*tan(1/2)^6 + 9*tan(1/2)^2) + 12*tan(1/2)^2*(pi*i + log(1 - tan(1/2)))/(-3 - 9*tan(1/2)^4 + 3*tan(1/2)^6 + 9*tan(1/2)^2) + 12*tan(1/2)^2*log(1 + tan(1/2))/(-3 - 9*tan(1/2)^4 + 3*tan(1/2)^6 + 9*tan(1/2)^2) + 12*tan(1/2)^4*log(1 + tan(1/2)^2)/(-3 - 9*tan(1/2)^4 + 3*tan(1/2)^6 + 9*tan(1/2)^2)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.