Métodos de integración : Sustitución trigonométrica - Integración producto de seno y coseno para resolverlos?

Por sustitucion trigonometrica

1)

<img src="https://tex.z-dn.net/?f=%5Cint%20%5Cfrac%7Bx%5E2%7D%7B%5Csqrt%7B25-x%5E2%7D%7Ddx...................%5C%3Ax%3D5%5Csin%20%5Cleft%28u%5Cright%29......dx%3D5%5Ccos%20%5Cleft%28u%5Cright%29du" />

reemplazando

<img src="https://tex.z-dn.net/?f=%3D%5Cint%20%5Cfrac%7B%5Cleft%285%5Csin%20%5Cleft%28u%5Cright%29%5Cright%29%5E2%7D%7B%5Csqrt%7B25-%5Cleft%285%5Csin%20%5Cleft%28u%5Cright%29%5Cright%29%5E2%7D%7D5%5Ccos%20%5Cleft%28u%5Cright%29du" />

<img src="https://tex.z-dn.net/?f=%3D125%5Cint%20%5Cfrac%7B%5Csin%20%5E2%5Cleft%28u%5Cright%29%5Ccos%20%5Cleft%28u%5Cright%29%7D%7B%5Csqrt%7B25-25%5Csin%20%5E2%5Cleft%28u%5Cright%29%7D%7Ddu%3D125%5Cint%20%5Cfrac%7B%5Csin%20%5E2%5Cleft%28u%5Cright%29%5Ccos%20%5Cleft%28u%5Cright%29%7D%7B5%5Csqrt%7B-%5Csin%20%5E2%5Cleft%28u%5Cright%29%2B1%7D%7Ddu" />

usamos lapitagorica para el denominador

<img src="https://tex.z-dn.net/?f=%5C%3A1-%5Csin%20%5E2%5Cleft%28x%5Cright%29%3D%5Ccos%20%5E2%5Cleft%28x%5Cright%29" />

<img src="https://tex.z-dn.net/?f=%3D%20%5Cfrac%7B125%7D%7B5%7D%20%5Cint%20%5Cfrac%7B%5Ccos%20%5Cleft%28u%5Cright%29%5Csin%20%5E2%5Cleft%28u%5Cright%29%7D%7B%5Csqrt%7B%5Ccos%20%5E2%5Cleft%28u%5Cright%29%7D%7Ddu%3D25%5Cint%20%5Csin%20%5E2%5Cleft%28u%5Cright%29du" />

con la identidad

<img src="https://tex.z-dn.net/?f=%5Csin%20%5E2%5Cleft%28x%5Cright%29%3D%5Cfrac%7B1-%5Ccos%20%5Cleft%282x%5Cright%29%7D%7B2%7D" />

<img src="https://tex.z-dn.net/?f=%3D25%5Cint%20%5Cfrac%7B1-%5Ccos%20%5Cleft%282u%5Cright%29%7D%7B2%7Ddu%3D%20%5Cfrac%7B25%7D%7B2%7D%28%5Cint%20%5C%3A1du-%5Cint%20%5Ccos%20%5Cleft%282u%5Cright%29du%29%3D%5Cfrac%7B25%7D%7B2%7D%5Cleft%28u-%5Cfrac%7B1%7D%7B2%7D%5Csin%20%5Cleft%282u%5Cright%29%5Cright%29%20" />

<img src="https://tex.z-dn.net/?f=x%3D5sen%5E2%28u%29.............u%3D%5Carcsin%20%5Cleft%28%5Cfrac%7B1%7D%7B5%7Dx%5Cright%29" />

<img src="https://tex.z-dn.net/?f=%3D%5Cfrac%7B25%7D%7B2%7D%5Cleft%28%5Carcsin%20%5Cleft%28%5Cfrac%7Bx%7D%7B5%7D%5Cright%29-%5Cfrac%7B1%7D%7B2%7D%5Csin%20%5Cleft%282%5Carcsin%20%5Cleft%28%5Cfrac%7Bx%7D%7B5%7D%5Cright%29%5Cright%29%5Cright%29%2Bk" />

2)

<img src="https://tex.z-dn.net/?f=%5Cint%20%5Cfrac%7B1%7D%7B%5Cleft%281%2Bx%5E2%5Cright%29%5E%7B%5Cfrac%7B3%7D%7B2%7D%7D%7Ddx...........x%3Dtan%28u%29....%5C%3Adx%3D%5Csec%20%5E2%5Cleft%28u%5Cright%29du" />

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como

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4)[img = 15]

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