Sr Examen
Integral de (7*cos(2x))/(cos(x)+sin(x)) dx
Solución
Ha introducido
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1 / | | 7*cos(2*x) | --------------- dx | cos(x) + sin(x) | / 0
$$\int\limits_{0}^{1} \frac{7 \cos{\left(2 x \right)}}{\sin{\left(x \right)} + \cos{\left(x \right)}}\, dx$$
Integral((7*cos(2*x))/(cos(x) + sin(x)), (x, 0, 1))
Respuesta (Indefinida)
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/ /x\ ___ / ___ /x\\ ___ / ___ /x\\ ___ / ___ /x\\ ___ / ___ /x\\ ___ 2/x\ / ___ /x\\ ___ 2/x\ / ___ /x\\ | 56*tan|-| 7*\/ 2 *log|-1 + \/ 2 + tan|-|| 7*\/ 2 *log|-1 - \/ 2 + tan|-|| 14*\/ 2 *log|-1 - \/ 2 + tan|-|| 14*\/ 2 *log|-1 + \/ 2 + tan|-|| 14*\/ 2 *tan |-|*log|-1 - \/ 2 + tan|-|| 14*\/ 2 *tan |-|*log|-1 + \/ 2 + tan|-|| | 7*cos(2*x) 56 \2/ \ \2// \ \2// \ \2// \ \2// \2/ \ \2// \2/ \ \2// | --------------- dx = C + ------------- + ------------- - -------------------------------- + -------------------------------- - --------------------------------- + --------------------------------- - ----------------------------------------- + ----------------------------------------- | cos(x) + sin(x) 2/x\ 2/x\ 2 2 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/
$$\int \frac{7 \cos{\left(2 x \right)}}{\sin{\left(x \right)} + \cos{\left(x \right)}}\, dx = C - \frac{7 \sqrt{2} \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 + \sqrt{2} \right)}}{2} + \frac{7 \sqrt{2} \log{\left(\tan{\left(\frac{x}{2} \right)} - \sqrt{2} - 1 \right)}}{2} + \frac{14 \sqrt{2} \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 + \sqrt{2} \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{4 \tan^{2}{\left(\frac{x}{2} \right)} + 4} + \frac{14 \sqrt{2} \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 + \sqrt{2} \right)}}{4 \tan^{2}{\left(\frac{x}{2} \right)} + 4} - \frac{14 \sqrt{2} \log{\left(\tan{\left(\frac{x}{2} \right)} - \sqrt{2} - 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{4 \tan^{2}{\left(\frac{x}{2} \right)} + 4} - \frac{14 \sqrt{2} \log{\left(\tan{\left(\frac{x}{2} \right)} - \sqrt{2} - 1 \right)}}{4 \tan^{2}{\left(\frac{x}{2} \right)} + 4} + \frac{56 \tan{\left(\frac{x}{2} \right)}}{4 \tan^{2}{\left(\frac{x}{2} \right)} + 4} + \frac{56}{4 \tan^{2}{\left(\frac{x}{2} \right)} + 4}$$
Gráfica
Respuesta
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___ / ___ \ ___ / / ___ \\ ___ / / ___ \\ ___ / ___ \ ___ 2 / / ___ \\ ___ 2 / ___ \
56 56*tan(1/2) 7*\/ 2 *log\-1 + \/ 2 + tan(1/2)/ 7*\/ 2 *\pi*I + log\1 + \/ 2 - tan(1/2)// 14*\/ 2 *\pi*I + log\1 + \/ 2 - tan(1/2)// 14*\/ 2 *log\-1 + \/ 2 + tan(1/2)/ 14*\/ 2 *tan (1/2)*\pi*I + log\1 + \/ 2 - tan(1/2)// 14*\/ 2 *tan (1/2)*log\-1 + \/ 2 + tan(1/2)/
-14 + --------------- + --------------- - ---------------------------------- + ------------------------------------------ - ------------------------------------------- + ----------------------------------- - ----------------------------------------------------- + ---------------------------------------------
2 2 2 2 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)
$$-14 + \frac{14 \sqrt{2} \log{\left(-1 + \tan{\left(\frac{1}{2} \right)} + \sqrt{2} \right)}}{4 \tan^{2}{\left(\frac{1}{2} \right)} + 4} + \frac{14 \sqrt{2} \log{\left(-1 + \tan{\left(\frac{1}{2} \right)} + \sqrt{2} \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{4 \tan^{2}{\left(\frac{1}{2} \right)} + 4} - \frac{7 \sqrt{2} \log{\left(-1 + \tan{\left(\frac{1}{2} \right)} + \sqrt{2} \right)}}{2} + \frac{56 \tan{\left(\frac{1}{2} \right)}}{4 \tan^{2}{\left(\frac{1}{2} \right)} + 4} + \frac{56}{4 \tan^{2}{\left(\frac{1}{2} \right)} + 4} - \frac{14 \sqrt{2} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + 1 + \sqrt{2} \right)} + i \pi\right)}{4 \tan^{2}{\left(\frac{1}{2} \right)} + 4} - \frac{14 \sqrt{2} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + 1 + \sqrt{2} \right)} + i \pi\right) \tan^{2}{\left(\frac{1}{2} \right)}}{4 \tan^{2}{\left(\frac{1}{2} \right)} + 4} + \frac{7 \sqrt{2} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + 1 + \sqrt{2} \right)} + i \pi\right)}{2}$$
=
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___ / ___ \ ___ / / ___ \\ ___ / / ___ \\ ___ / ___ \ ___ 2 / / ___ \\ ___ 2 / ___ \
56 56*tan(1/2) 7*\/ 2 *log\-1 + \/ 2 + tan(1/2)/ 7*\/ 2 *\pi*I + log\1 + \/ 2 - tan(1/2)// 14*\/ 2 *\pi*I + log\1 + \/ 2 - tan(1/2)// 14*\/ 2 *log\-1 + \/ 2 + tan(1/2)/ 14*\/ 2 *tan (1/2)*\pi*I + log\1 + \/ 2 - tan(1/2)// 14*\/ 2 *tan (1/2)*log\-1 + \/ 2 + tan(1/2)/
-14 + --------------- + --------------- - ---------------------------------- + ------------------------------------------ - ------------------------------------------- + ----------------------------------- - ----------------------------------------------------- + ---------------------------------------------
2 2 2 2 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)
$$-14 + \frac{14 \sqrt{2} \log{\left(-1 + \tan{\left(\frac{1}{2} \right)} + \sqrt{2} \right)}}{4 \tan^{2}{\left(\frac{1}{2} \right)} + 4} + \frac{14 \sqrt{2} \log{\left(-1 + \tan{\left(\frac{1}{2} \right)} + \sqrt{2} \right)} \tan^{2}{\left(\frac{1}{2} \right)}}{4 \tan^{2}{\left(\frac{1}{2} \right)} + 4} - \frac{7 \sqrt{2} \log{\left(-1 + \tan{\left(\frac{1}{2} \right)} + \sqrt{2} \right)}}{2} + \frac{56 \tan{\left(\frac{1}{2} \right)}}{4 \tan^{2}{\left(\frac{1}{2} \right)} + 4} + \frac{56}{4 \tan^{2}{\left(\frac{1}{2} \right)} + 4} - \frac{14 \sqrt{2} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + 1 + \sqrt{2} \right)} + i \pi\right)}{4 \tan^{2}{\left(\frac{1}{2} \right)} + 4} - \frac{14 \sqrt{2} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + 1 + \sqrt{2} \right)} + i \pi\right) \tan^{2}{\left(\frac{1}{2} \right)}}{4 \tan^{2}{\left(\frac{1}{2} \right)} + 4} + \frac{7 \sqrt{2} \left(\log{\left(- \tan{\left(\frac{1}{2} \right)} + 1 + \sqrt{2} \right)} + i \pi\right)}{2}$$
-14 + 56/(4 + 4*tan(1/2)^2) + 56*tan(1/2)/(4 + 4*tan(1/2)^2) - 7*sqrt(2)*log(-1 + sqrt(2) + tan(1/2))/2 + 7*sqrt(2)*(pi*i + log(1 + sqrt(2) - tan(1/2)))/2 - 14*sqrt(2)*(pi*i + log(1 + sqrt(2) - tan(1/2)))/(4 + 4*tan(1/2)^2) + 14*sqrt(2)*log(-1 + sqrt(2) + tan(1/2))/(4 + 4*tan(1/2)^2) - 14*sqrt(2)*tan(1/2)^2*(pi*i + log(1 + sqrt(2) - tan(1/2)))/(4 + 4*tan(1/2)^2) + 14*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.