What is the integral of cos(x)(1+5sin^2(x)) ?

The integral of cos(x)(1+5sin^2(x)) is sin(x)+(5sin^3(x))/3+C

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integral of cos(x)(1+5sin^2(x))

\int cos (x) (1 + 5 sin^{2} (x)) d x

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

Solution steps

\int cos (x) (1 + 5 sin^{2} (x)) d x

Apply u-substitution

: \int 1 + 5 u^{2} d u

= \int 1 + 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 1 d u + \int 5 u^{2} d u

\int 1 d u = u

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

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

Substitute back

u = sin (x)

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

Add a constant to the solution

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

n = 0 \sum \infty \frac{1}{n ^{2} + n + 1}

x \to 0 lim (ln (x) + \frac{1}{x})

\int 2 e^{- 2 x} d x

\frac{x d y}{d x} = - y + 2 x^{2} y

\int \frac{5 x - 7 x ^{2} + 4}{( x - 2 ) ^{2} ( x + 1 )} d x

What is the integral of cos(x)(1+5sin^2(x)) ?
The integral of cos(x)(1+5sin^2(x)) is sin(x)+(5sin^3(x))/3+C