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

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

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

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