What is the integral of cos(x)(6+8sin^2(x)) ?

The integral of cos(x)(6+8sin^2(x)) is 6sin(x)+(8sin^3(x))/3+C

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integral of cos(x)(6+8sin^2(x))

\int cos (x) (6 + 8 sin^{2} (x)) d x

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

Solution steps

\int cos (x) (6 + 8 sin^{2} (x)) d x

Apply u-substitution

: \int 6 + 8 u^{2} d u

= \int 6 + 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 6 d u + \int 8 u^{2} d u

\int 6 d u = 6 u

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

= 6 u + \frac{8 u ^{3}}{3}

Substitute back

u = sin (x)

= 6 sin (x) + \frac{8 sin ^{3} ( x )}{3}

Add a constant to the solution

= 6 sin (x) + \frac{8 sin ^{3} ( x )}{3} + C

2 x^{2}, 8 x^{2}, - 3 x + 5

\int \frac{1}{sin ^{2} ( x ) cos ^{4} ( x )} d x

\frac{d}{d x} (e^{x} sin (x) + cos (x))

x \to - 5 lim ((5 + x)^{2} - 25)

cos (4 - 9 x^{2})

What is the integral of cos(x)(6+8sin^2(x)) ?
The integral of cos(x)(6+8sin^2(x)) is 6sin(x)+(8sin^3(x))/3+C