What is the integral of cos(x)(7+8sin^2(x)) ?

The integral of cos(x)(7+8sin^2(x)) is 7sin(x)+(8sin^3(x))/3+C

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integral of cos(x)(7+8sin^2(x))

\int cos (x) (7 + 8 sin^{2} (x)) d x

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

Solution steps

\int cos (x) (7 + 8 sin^{2} (x)) d x

Apply u-substitution

= \int 7 + 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 7 d u + \int 8 u^{2} d u

\int 7 d u = 7 u

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

= 7 u + \frac{8 u ^{3}}{3}

Substitute back

u = sin (x)

= 7 sin (x) + \frac{8 sin ^{3} ( x )}{3}

Add a constant to the solution

= 7 sin (x) + \frac{8 sin ^{3} ( x )}{3} + C

6 y^{^{''}} + 40 y^{^{'}} + 54 y = 0

\frac{d}{d x} (- k x)

\int cos (x) - x sin (x) d x

e^{- x^{62}}

y^{^{'}} = x - 2 y

What is the integral of cos(x)(7+8sin^2(x)) ?
The integral of cos(x)(7+8sin^2(x)) is 7sin(x)+(8sin^3(x))/3+C