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