ALGUIEN SABE COMO SE HACE ESTA INTEGRAL DOY 40 PUNTOS : [tex] \ int _{ \ frac{ \ pi }{6 \ : }} ^ { \ frac{ \ pi }{4} \ : } \ : \ frac{sin \ left(x \ right)dx}{cos ^ 3 \ left(x \ right) - sin ^ 3 \ left(x \ right)}[ / tex].
En resumen
Hagamos la sustitución : <img src="https://tex.z-dn.net/?f=x%3D%5Carctan%20u" /> entonces <img src="https://tex.z-dn.net/?
Hagamos la sustitución : <img src="https://tex.z-dn.net/?f=x%3D%5Carctan%20u" /> entonces <img src="https://tex.z-dn.net/?f=dx%20%3D%20%5Cdfrac%7B1%7D%7B1%2Bu%5E2%7D%5C%2Cdu%5C%5C%20%5C%5C%20%5C%5C%0A%5Csin%20x%20%3D%5Cdfrac%7Bu%7D%7B%5Csqrt%7B1%2Bu%5E2%7D%7D%5C%5C%20%5C%5C%20%5C%5C%0A%5Ccos%20x%20%3D%20%5Cdfrac%7B1%7D%7B%5Csqrt%7B1%2Bu%5E2%7D%7D" />
Sustituyamos
<img src="https://tex.z-dn.net/?f=%5Cdisplaystyle%0AI%3D%5Cint_%7B%5Ctan%20%28%5Cpi%2F6%29%7D%5E%7B%5Ctan%28%5Cpi%2F4%29%7D%20%5Cdfrac%7Bu%7D%7B1-u%5E3%7D%5C%2Cdu%5C%5C%20%5C%5C%20%5C%5C%0AI%3D%5Cint_%7B1%2F%5Csqrt%7B3%7D%7D%5E%7B1%2F%5Csqrt%7B2%7D%7D%20%5Cdfrac%7Bu%7D%7B3%28u%5E2%2Bu%2B1%29%7D-%5Cdfrac%7B1%7D%7B3%28u%5E2%2Bu%2B1%29%7D-%5Cdfrac%7B1%7D%7B3%28u-1%29%7D%0A" />
<img src="https://tex.z-dn.net/?f=%5Cdisplaystyle%0AI%3D%5Cdfrac%7B1%7D%7B6%7D%5Cint_%7B1%2F%5Csqrt%7B3%7D%7D%5E%7B1%2F%5Csqrt%7B2%7D%7D%5Cdfrac%7B2u%2B1%7D%7Bu%5E2%2Bu%2B1%7D-%5Cdfrac%7B3%7D%7Bu%5E2%2Bu%2B1%7D%5C%2C%20du-%5Cdfrac%7B1%7D%7B3%7D%5Cleft.%5Cleft%28%5Cln%20%7Cu-1%7C%5Cright%29%5Cright%7C_%7B1%2F%5Csqrt%7B3%7D%7D%5E%7B1%2F%5Csqrt%7B2%7D%7D%5C%5C%20%5C%5C%20%5C%5C%0AI%3D%5Cdfrac%7B1%7D%7B6%7D%5Cleft.%5Cleft%5B%5Cln%20%28u%5E2%2Bu%2B1%29%5Cright%5D%5Cright%7C_%7B1%2F%5Csqrt%7B3%7D%7D%5E%7B1%2F%5Csqrt%7B2%7D%7D-%5Cdfrac%7B1%7D%7B2%7D%5Cint_%7B1%2F%5Csqrt%7B3%7D%7D%5E%7B1%2F%5Csqrt%7B2%7D%7D%5Cdfrac%7Bdu%7D%7B%5Cleft%28u%2B%5Cdfrac%7B1%7D%7B2%7D%5Cright%29%5E2%2B%5Cdfrac%7B3%7D%7B4%7D%7D-%5Ccdots%5C%5C%20%5C%5C%0A%5Ccdots-%5Cdfrac%7B1%7D%7B3%7D%5Cln%20%5Cleft%7C%5Cdfrac%7B%5Csqrt%7B6%7D-%5Csqrt%7B3%7D%7D%7B%5Csqrt%7B6%7D-%5Csqrt%7B2%7D%7D%5Cright%7C%5C%5C%20%5C%5C%20%5C%5C%0A" />
<img src="https://tex.z-dn.net/?f=I%3D%5Cdfrac%7B1%7D%7B6%7D%5Cln%5Cleft%28%5Cdfrac%7B9%2B3%5Csqrt%7B2%7D%7D%7B8%2B2%5Csqrt%7B3%7D%7D%5Cright%29-%5Cdfrac%7B1%7D%7B%5Csqrt%7B3%7D%7D%5Cleft%5B%5Carctan%5Cleft%28%5Cdfrac%7B2u%2B1%7D%7B%5Csqrt%7B3%7D%7D%5Cright%29%5Cright%5D_%7B1%2F%5Csqrt%7B3%7D%7D%5E%7B1%2F%5Csqrt%7B2%7D%7D-%5Cdfrac%7B1%7D%7B3%7D%5Cln%5Cleft%28%5Cdfrac%7B%5Csqrt%7B6%7D-%5Csqrt%7B3%7D%7D%7B%5Csqrt%7B6%7D-%5Csqrt%7B2%7D%7D%5Cright%29%5C%5C%20%5C%5C%20%5C%5C%0A" />
De allí continua con el arcotangente.
La resuelves con sustitución simple. Sea u = x² + 3 du = 2x dx du / 2 = x dx Sustituyedo en términos de u 4∫(1 / u)(du / 2) = (4 / 2)∫(du / u) integrando : = 2 ln(u) + c volviendo a la variable original u = x² + 3 =…
Te envio algo para que lo revises.
Mira el archivo adjunto.
.