Sr Examen
Integral de (-4(cos(x))^2*sin(x))/(cos(x)+sin(x)) dx
Solución
Ha introducido
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1 / | | 2 | -4*cos (x)*sin(x) | ----------------- dx | cos(x) + sin(x) | / 0
$$\int\limits_{0}^{1} \frac{\sin{\left(x \right)} \left(- 4 \cos^{2}{\left(x \right)}\right)}{\sin{\left(x \right)} + \cos{\left(x \right)}}\, dx$$
Integral(((-4*cos(x)^2)*sin(x))/(cos(x) + sin(x)), (x, 0, 1))
Respuesta (Indefinida)
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/ | 2/x\ /x\ / 2/x\\ / 2/x\ /x\\ 3/x\ 2/x\ / 2/x\\ 4/x\ / 2/x\\ 4/x\ / 2/x\ /x\\ 2/x\ / 2/x\ /x\\ | 2 16*tan |-| 8*tan|-| 4*log|1 + tan |-|| 4*log|1 - tan |-| + 2*tan|-|| 8*tan |-| 8*tan |-|*log|1 + tan |-|| 4*tan |-|*log|1 + tan |-|| 4*tan |-|*log|1 - tan |-| + 2*tan|-|| 8*tan |-|*log|1 - tan |-| + 2*tan|-|| | -4*cos (x)*sin(x) \2/ \2/ \ \2// \ \2/ \2// \2/ \2/ \ \2// \2/ \ \2// \2/ \ \2/ \2// \2/ \ \2/ \2// | ----------------- dx = C - ------------------------- - ------------------------- - ------------------------- + ----------------------------- + ------------------------- - -------------------------- - -------------------------- + ------------------------------------- + ------------------------------------- | cos(x) + sin(x) 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ 4/x\ 2/x\ | 4 + 4*tan |-| + 8*tan |-| 4 + 4*tan |-| + 8*tan |-| 4 + 4*tan |-| + 8*tan |-| 4 + 4*tan |-| + 8*tan |-| 4 + 4*tan |-| + 8*tan |-| 4 + 4*tan |-| + 8*tan |-| 4 + 4*tan |-| + 8*tan |-| 4 + 4*tan |-| + 8*tan |-| 4 + 4*tan |-| + 8*tan |-| / \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/ \2/
$$\int \frac{\sin{\left(x \right)} \left(- 4 \cos^{2}{\left(x \right)}\right)}{\sin{\left(x \right)} + \cos{\left(x \right)}}\, dx = C - \frac{4 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{4}{\left(\frac{x}{2} \right)}}{4 \tan^{4}{\left(\frac{x}{2} \right)} + 8 \tan^{2}{\left(\frac{x}{2} \right)} + 4} - \frac{8 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{4 \tan^{4}{\left(\frac{x}{2} \right)} + 8 \tan^{2}{\left(\frac{x}{2} \right)} + 4} - \frac{4 \log{\left(\tan^{2}{\left(\frac{x}{2} \right)} + 1 \right)}}{4 \tan^{4}{\left(\frac{x}{2} \right)} + 8 \tan^{2}{\left(\frac{x}{2} \right)} + 4} + \frac{4 \log{\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 2 \tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{4}{\left(\frac{x}{2} \right)}}{4 \tan^{4}{\left(\frac{x}{2} \right)} + 8 \tan^{2}{\left(\frac{x}{2} \right)} + 4} + \frac{8 \log{\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 2 \tan{\left(\frac{x}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{4 \tan^{4}{\left(\frac{x}{2} \right)} + 8 \tan^{2}{\left(\frac{x}{2} \right)} + 4} + \frac{4 \log{\left(- \tan^{2}{\left(\frac{x}{2} \right)} + 2 \tan{\left(\frac{x}{2} \right)} + 1 \right)}}{4 \tan^{4}{\left(\frac{x}{2} \right)} + 8 \tan^{2}{\left(\frac{x}{2} \right)} + 4} + \frac{8 \tan^{3}{\left(\frac{x}{2} \right)}}{4 \tan^{4}{\left(\frac{x}{2} \right)} + 8 \tan^{2}{\left(\frac{x}{2} \right)} + 4} - \frac{16 \tan^{2}{\left(\frac{x}{2} \right)}}{4 \tan^{4}{\left(\frac{x}{2} \right)} + 8 \tan^{2}{\left(\frac{x}{2} \right)} + 4} - \frac{8 \tan{\left(\frac{x}{2} \right)}}{4 \tan^{4}{\left(\frac{x}{2} \right)} + 8 \tan^{2}{\left(\frac{x}{2} \right)} + 4}$$
Gráfica
Respuesta
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2 / 2 \ / 2 \ 3 2 / 2 \ 4 / 2 \ 4 / 2 \ 2 / 2 \
16*tan (1/2) 8*tan(1/2) 4*log\1 + tan (1/2)/ 4*log\1 - tan (1/2) + 2*tan(1/2)/ 8*tan (1/2) 8*tan (1/2)*log\1 + tan (1/2)/ 4*tan (1/2)*log\1 + tan (1/2)/ 4*tan (1/2)*log\1 - tan (1/2) + 2*tan(1/2)/ 8*tan (1/2)*log\1 - tan (1/2) + 2*tan(1/2)/
- ----------------------------- - ----------------------------- - ----------------------------- + --------------------------------- + ----------------------------- - ------------------------------ - ------------------------------ + ------------------------------------------- + -------------------------------------------
4 2 4 2 4 2 4 2 4 2 4 2 4 2 4 2 4 2
4 + 4*tan (1/2) + 8*tan (1/2) 4 + 4*tan (1/2) + 8*tan (1/2) 4 + 4*tan (1/2) + 8*tan (1/2) 4 + 4*tan (1/2) + 8*tan (1/2) 4 + 4*tan (1/2) + 8*tan (1/2) 4 + 4*tan (1/2) + 8*tan (1/2) 4 + 4*tan (1/2) + 8*tan (1/2) 4 + 4*tan (1/2) + 8*tan (1/2) 4 + 4*tan (1/2) + 8*tan (1/2)
$$- \frac{16 \tan^{2}{\left(\frac{1}{2} \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4} - \frac{8 \tan{\left(\frac{1}{2} \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4} - \frac{4 \log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4} - \frac{8 \log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4} - \frac{4 \log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{4}{\left(\frac{1}{2} \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4} + \frac{4 \log{\left(- \tan^{2}{\left(\frac{1}{2} \right)} + 1 + 2 \tan{\left(\frac{1}{2} \right)} \right)} \tan^{4}{\left(\frac{1}{2} \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4} + \frac{8 \tan^{3}{\left(\frac{1}{2} \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4} + \frac{8 \log{\left(- \tan^{2}{\left(\frac{1}{2} \right)} + 1 + 2 \tan{\left(\frac{1}{2} \right)} \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4} + \frac{4 \log{\left(- \tan^{2}{\left(\frac{1}{2} \right)} + 1 + 2 \tan{\left(\frac{1}{2} \right)} \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4}$$
=
=
2 / 2 \ / 2 \ 3 2 / 2 \ 4 / 2 \ 4 / 2 \ 2 / 2 \
16*tan (1/2) 8*tan(1/2) 4*log\1 + tan (1/2)/ 4*log\1 - tan (1/2) + 2*tan(1/2)/ 8*tan (1/2) 8*tan (1/2)*log\1 + tan (1/2)/ 4*tan (1/2)*log\1 + tan (1/2)/ 4*tan (1/2)*log\1 - tan (1/2) + 2*tan(1/2)/ 8*tan (1/2)*log\1 - tan (1/2) + 2*tan(1/2)/
- ----------------------------- - ----------------------------- - ----------------------------- + --------------------------------- + ----------------------------- - ------------------------------ - ------------------------------ + ------------------------------------------- + -------------------------------------------
4 2 4 2 4 2 4 2 4 2 4 2 4 2 4 2 4 2
4 + 4*tan (1/2) + 8*tan (1/2) 4 + 4*tan (1/2) + 8*tan (1/2) 4 + 4*tan (1/2) + 8*tan (1/2) 4 + 4*tan (1/2) + 8*tan (1/2) 4 + 4*tan (1/2) + 8*tan (1/2) 4 + 4*tan (1/2) + 8*tan (1/2) 4 + 4*tan (1/2) + 8*tan (1/2) 4 + 4*tan (1/2) + 8*tan (1/2) 4 + 4*tan (1/2) + 8*tan (1/2)
$$- \frac{16 \tan^{2}{\left(\frac{1}{2} \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4} - \frac{8 \tan{\left(\frac{1}{2} \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4} - \frac{4 \log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4} - \frac{8 \log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4} - \frac{4 \log{\left(\tan^{2}{\left(\frac{1}{2} \right)} + 1 \right)} \tan^{4}{\left(\frac{1}{2} \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4} + \frac{4 \log{\left(- \tan^{2}{\left(\frac{1}{2} \right)} + 1 + 2 \tan{\left(\frac{1}{2} \right)} \right)} \tan^{4}{\left(\frac{1}{2} \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4} + \frac{8 \tan^{3}{\left(\frac{1}{2} \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4} + \frac{8 \log{\left(- \tan^{2}{\left(\frac{1}{2} \right)} + 1 + 2 \tan{\left(\frac{1}{2} \right)} \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4} + \frac{4 \log{\left(- \tan^{2}{\left(\frac{1}{2} \right)} + 1 + 2 \tan{\left(\frac{1}{2} \right)} \right)}}{4 \tan^{4}{\left(\frac{1}{2} \right)} + 8 \tan^{2}{\left(\frac{1}{2} \right)} + 4}$$
-16*tan(1/2)^2/(4 + 4*tan(1/2)^4 + 8*tan(1/2)^2) - 8*tan(1/2)/(4 + 4*tan(1/2)^4 + 8*tan(1/2)^2) - 4*log(1 + tan(1/2)^2)/(4 + 4*tan(1/2)^4 + 8*tan(1/2)^2) + 4*log(1 - tan(1/2)^2 + 2*tan(1/2))/(4 + 4*tan(1/2)^4 + 8*tan(1/2)^2) + 8*tan(1/2)^3/(4 + 4*tan(1/2)^4 + 8*tan(1/2)^2) - 8*tan(1/2)^2*log(1 + tan(1/2)^2)/(4 + 4*tan(1/2)^4 + 8*tan(1/2)^2) - 4*tan(1/2)^4*log(1 + tan(1/2)^2)/(4 + 4*tan(1/2)^4 + 8*tan(1/2)^2) + 4*tan(1/2)^4*log(1 - tan(1/2)^2 + 2*tan(1/2))/(4 + 4*tan(1/2)^4 + 8*tan(1/2)^2) + 8*tan(1/2)^2*log(1 - tan(1/2)^2 + 2*tan(1/2))/(4 + 4*tan(1/2)^4 + 8*tan(1/2)^2)
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.