What is the integral of cos(x)(8+5sin^2(x)) ?

The integral of cos(x)(8+5sin^2(x)) is 8sin(x)+(5sin^3(x))/3+C

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integral of cos(x)(8+5sin^2(x))

\int cos (x) (8 + 5 sin^{2} (x)) d x

8 sin (x) + \frac{5 sin ^{3} ( x )}{3} + C

Solution steps

\int cos (x) (8 + 5 sin^{2} (x)) d x

Apply u-substitution

: \int 8 + 5 u^{2} d u

= \int 8 + 5 u^{2} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 8 d u + \int 5 u^{2} d u

\int 8 d u = 8 u

\int 5 u^{2} d u = \frac{5 u ^{3}}{3}

= 8 u + \frac{5 u ^{3}}{3}

Substitute back

u = sin (x)

= 8 sin (x) + \frac{5 sin ^{3} ( x )}{3}

Add a constant to the solution

= 8 sin (x) + \frac{5 sin ^{3} ( x )}{3} + C

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\int_{1}^{\infty} \frac{1}{x ^{2} + 2} d x

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What is the integral of cos(x)(8+5sin^2(x)) ?
The integral of cos(x)(8+5sin^2(x)) is 8sin(x)+(5sin^3(x))/3+C