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

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

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

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

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

Solution steps

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

Apply u-substitution

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

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

\int 5 d u = 5 u

\int 8 u^{2} d u = \frac{8 u ^{3}}{3}

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

Substitute back

u = sin (x)

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

Add a constant to the solution

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

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

x, x^{2} - 1, 0, 1

6 x^{4} - 6 x^{2}, 5 x^{2}, [0, \frac{11}{6}]

\int \frac{6}{3 x - 2} d x

\frac{\partial}{\partial y} (x^{2} + y^{2} - 4)

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