Sr Examen
Integral de sin2x/(cosx+sinx) dx
Solución
Ha introducido
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pi -- 2 / | | sin(2*x) | --------------- dx | cos(x) + sin(x) | / 0
$$\int\limits_{0}^{\frac{\pi}{2}} \frac{\sin{\left(2 x \right)}}{\sin{\left(x \right)} + \cos{\left(x \right)}}\, dx$$
Integral(sin(2*x)/(cos(x) + sin(x)), (x, 0, pi/2))
Respuesta (Indefinida)
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/ /x\ ___ / ___ /x\\ ___ / ___ /x\\ ___ 2/x\ / ___ /x\\ ___ 2/x\ / ___ /x\\ | 8*tan|-| 2*\/ 2 *log|-1 + \/ 2 + tan|-|| 2*\/ 2 *log|-1 - \/ 2 + tan|-|| 2*\/ 2 *tan |-|*log|-1 + \/ 2 + tan|-|| 2*\/ 2 *tan |-|*log|-1 - \/ 2 + tan|-|| | sin(2*x) 8 \2/ \ \2// \ \2// \2/ \ \2// \2/ \ \2// | --------------- dx = C - ------------- + ------------- - -------------------------------- + -------------------------------- - ---------------------------------------- + ---------------------------------------- | cos(x) + sin(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/
$$\int \frac{\sin{\left(2 x \right)}}{\sin{\left(x \right)} + \cos{\left(x \right)}}\, dx = C - \frac{2 \sqrt{2} \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 + \sqrt{2} \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{4 \tan^{2}{\left(\frac{x}{2} \right)} + 4} - \frac{2 \sqrt{2} \log{\left(\tan{\left(\frac{x}{2} \right)} - 1 + \sqrt{2} \right)}}{4 \tan^{2}{\left(\frac{x}{2} \right)} + 4} + \frac{2 \sqrt{2} \log{\left(\tan{\left(\frac{x}{2} \right)} - \sqrt{2} - 1 \right)} \tan^{2}{\left(\frac{x}{2} \right)}}{4 \tan^{2}{\left(\frac{x}{2} \right)} + 4} + \frac{2 \sqrt{2} \log{\left(\tan{\left(\frac{x}{2} \right)} - \sqrt{2} - 1 \right)}}{4 \tan^{2}{\left(\frac{x}{2} \right)} + 4} + \frac{8 \tan{\left(\frac{x}{2} \right)}}{4 \tan^{2}{\left(\frac{x}{2} \right)} + 4} - \frac{8}{4 \tan^{2}{\left(\frac{x}{2} \right)} + 4}$$
Gráfica
Estos ejemplos se pueden aplicar para introducción de los límites de integración inferior y superior.