Question

Evaluate the integral. (Use C for the constant of integration.)
cos(x) (8 + 7 sin^2(x)) dx

Answer 1

Answer:
`8 \sin x +7((\sin^3x)/(3))+C`

Explanation:

Consider
`\int \cos(x) (8+7 \sin^2(x)) \, dx`

Substitute t= sinx

then dt = cos x dx

`\int \cos(x) (8+7 \sin^2(x)) \, dx = \int (8+7t^2)dt\\\\ =8t+7((t^3)/(3))+C`

`[\int x^ndx=(x^(n+1))/(n+1)+C]`

`=8 \sin x +7((\sin^3x)/(3))+C`

Hence,
`\int \cos(x) (8+7 \sin^2(x)) \, dx=8 \sin x +7((\sin^3x)/(3))+C`