Sr Examen
Integral de x/cosx^4 dx
Solución
Ha introducido
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1 / | | x | ------- dx | 4 | cos (x) | / 0
$$\int\limits_{0}^{1} \frac{x}{\cos^{4}{\left(x \right)}}\, dx$$
Integral(x/cos(x)^4, (x, 0, 1))
Respuesta (Indefinida)
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/ 4/x\ / /x\\ / /x\\ 2/x\ / 2/x\\ 5/x\ /x\ 2/x\ / 2/x\\ 4/x\ / /x\\ 4/x\ / /x\\ 6/x\ / 2/x\\ 6/x\ / /x\\ 6/x\ / /x\\ 3/x\ 2/x\ / /x\\ 2/x\ / /x\\ 4/x\ / 2/x\\ | 2*tan |-| 2*log|1 + tan|-|| 2*log|-1 + tan|-|| 2*tan |-| 2*log|1 + tan |-|| 6*x*tan |-| 6*x*tan|-| 6*tan |-|*log|1 + tan |-|| 6*tan |-|*log|1 + tan|-|| 6*tan |-|*log|-1 + tan|-|| 2*tan |-|*log|1 + tan |-|| 2*tan |-|*log|1 + tan|-|| 2*tan |-|*log|-1 + tan|-|| 4*x*tan |-| 6*tan |-|*log|1 + tan|-|| 6*tan |-|*log|-1 + tan|-|| 6*tan |-|*log|1 + tan |-|| | x \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// | ------- dx = C - -------------------------------------- - -------------------------------------- - -------------------------------------- + -------------------------------------- + -------------------------------------- - -------------------------------------- - -------------------------------------- - -------------------------------------- - -------------------------------------- - -------------------------------------- - -------------------------------------- + -------------------------------------- + -------------------------------------- + -------------------------------------- + -------------------------------------- + -------------------------------------- + -------------------------------------- | 4 4/x\ 6/x\ 2/x\ 4/x\ 6/x\ 2/x\ 4/x\ 6/x\ 2/x\ 4/x\ 6/x\ 2/x\ 4/x\ 6/x\ 2/x\ 4/x\ 6/x\ 2/x\ 4/x\ 6/x\ 2/x\ 4/x\ 6/x\ 2/x\ 4/x\ 6/x\ 2/x\ 4/x\ 6/x\ 2/x\ 4/x\ 6/x\ 2/x\ 4/x\ 6/x\ 2/x\ 4/x\ 6/x\ 2/x\ 4/x\ 6/x\ 2/x\ 4/x\ 6/x\ 2/x\ 4/x\ 6/x\ 2/x\ 4/x\ 6/x\ 2/x\ | cos (x) -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{x}{\cos^{4}{\left(x \right)}}\, dx = C - \frac{6 x \tan^{5}{\left(\frac{x}{2} \right)}}{3 \tan^{6}{\left(\frac{x}{2} \right)} - 9 \tan^{4}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} - 3} + \frac{4 x \tan^{3}{\left(\frac{x}{2} \right)}}{3 \tan^{6}{\left(\frac{x}{2} \right)} - 9 \tan^{4}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} - 3} - \frac{6 x \tan{\left(\frac{x}{2} \right)}}{3 \tan^{6}{\left(\frac{x}{2} \right)} - 9 \tan^{4}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} - 3} + \frac{2 \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} \tan^{6}{\left(\frac{x}{2} \right)}}{3 \tan^{6}{\left(\frac{x}{2} \right)} - 9 \tan^{4}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} - 3} - \frac{6 \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} \tan^{4}{\left(\frac{x}{2} \right)}}{3 \tan^{6}{\left(\frac{x}{2} \right)} - 9 \tan^{4}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} - 3} + \frac{6 \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{3 \tan^{6}{\left(\frac{x}{2} \right)} - 9 \tan^{4}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} - 3} - \frac{2 \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 \right)}}{3 \tan^{6}{\left(\frac{x}{2} \right)} - 9 \tan^{4}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} - 3} + \frac{2 \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{6}{\left(\frac{x}{2} \right)}}{3 \tan^{6}{\left(\frac{x}{2} \right)} - 9 \tan^{4}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} - 3} - \frac{6 \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{4}{\left(\frac{x}{2} \right)}}{3 \tan^{6}{\left(\frac{x}{2} \right)} - 9 \tan^{4}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} - 3} + \frac{6 \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{3 \tan^{6}{\left(\frac{x}{2} \right)} - 9 \tan^{4}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} - 3} - \frac{2 \log{\left(\tan{\left(\frac{x}{2} \right)} + 1 \right)}}{3 \tan^{6}{\left(\frac{x}{2} \right)} - 9 \tan^{4}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} - 3} - \frac{2 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{6}{\left(\frac{x}{2} \right)}}{3 \tan^{6}{\left(\frac{x}{2} \right)} - 9 \tan^{4}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} - 3} + \frac{6 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{4}{\left(\frac{x}{2} \right)}}{3 \tan^{6}{\left(\frac{x}{2} \right)} - 9 \tan^{4}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} - 3} - \frac{6 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{3 \tan^{6}{\left(\frac{x}{2} \right)} - 9 \tan^{4}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} - 3} + \frac{2 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)}}{3 \tan^{6}{\left(\frac{x}{2} \right)} - 9 \tan^{4}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} - 3} - \frac{2 \tan^{4}{\left(\frac{x}{2} \right)}}{3 \tan^{6}{\left(\frac{x}{2} \right)} - 9 \tan^{4}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{2} \right)} - 3} + \frac{2 \tan^{2}{\left(\frac{x}{2} \right)}}{3 \tan^{6}{\left(\frac{x}{2} \right)} - 9 \tan^{4}{\left(\frac{x}{2} \right)} + 9 \tan^{2}{\left(\frac{x}{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 \
6*tan (1/2) 6*tan(1/2) 2*tan (1/2) 2*(pi*I + log(1 - tan(1/2))) 2*log(1 + tan(1/2)) 2*tan (1/2) 2*log\1 + tan (1/2)/ 4*tan (1/2) 2*pi*I 6*tan (1/2)*log\1 + tan (1/2)/ 6*tan (1/2)*(pi*I + log(1 - tan(1/2))) 6*tan (1/2)*log(1 + tan(1/2)) 2*tan (1/2)*log\1 + tan (1/2)/ 2*tan (1/2)*(pi*I + log(1 - tan(1/2))) 2*tan (1/2)*log(1 + tan(1/2)) 6*tan (1/2)*(pi*I + log(1 - tan(1/2))) 6*tan (1/2)*log(1 + tan(1/2)) 6*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{6 \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{4 \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{2 \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{2 \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{6 \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{2 \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{2 \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{2 \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{6 \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{6 \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{6 \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{2 \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{6 \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{6 \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{2 i \pi}{3} + \frac{2 \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{6 \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{2 \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 \
6*tan (1/2) 6*tan(1/2) 2*tan (1/2) 2*(pi*I + log(1 - tan(1/2))) 2*log(1 + tan(1/2)) 2*tan (1/2) 2*log\1 + tan (1/2)/ 4*tan (1/2) 2*pi*I 6*tan (1/2)*log\1 + tan (1/2)/ 6*tan (1/2)*(pi*I + log(1 - tan(1/2))) 6*tan (1/2)*log(1 + tan(1/2)) 2*tan (1/2)*log\1 + tan (1/2)/ 2*tan (1/2)*(pi*I + log(1 - tan(1/2))) 2*tan (1/2)*log(1 + tan(1/2)) 6*tan (1/2)*(pi*I + log(1 - tan(1/2))) 6*tan (1/2)*log(1 + tan(1/2)) 6*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{6 \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{4 \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{2 \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{2 \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{6 \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{2 \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{2 \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{2 \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{6 \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{6 \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{6 \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{2 \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{6 \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{6 \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{2 i \pi}{3} + \frac{2 \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{6 \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{2 \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)}}$$
-6*tan(1/2)^5/(-3 - 9*tan(1/2)^4 + 3*tan(1/2)^6 + 9*tan(1/2)^2) - 6*tan(1/2)/(-3 - 9*tan(1/2)^4 + 3*tan(1/2)^6 + 9*tan(1/2)^2) - 2*tan(1/2)^4/(-3 - 9*tan(1/2)^4 + 3*tan(1/2)^6 + 9*tan(1/2)^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) - 2*log(1 + tan(1/2))/(-3 - 9*tan(1/2)^4 + 3*tan(1/2)^6 + 9*tan(1/2)^2) + 2*tan(1/2)^2/(-3 - 9*tan(1/2)^4 + 3*tan(1/2)^6 + 9*tan(1/2)^2) + 2*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)^3/(-3 - 9*tan(1/2)^4 + 3*tan(1/2)^6 + 9*tan(1/2)^2) - 2*pi*i/3 - 6*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) - 6*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) - 6*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) - 2*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) + 2*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) + 2*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) + 6*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) + 6*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) + 6*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.