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