What is the integral of cos(x)(1+7sin^2(x)) ?

The integral of cos(x)(1+7sin^2(x)) is sin(x)+(7sin^3(x))/3+C

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integral of cos(x)(1+7sin^2(x))

\int cos (x) (1 + 7 sin^{2} (x)) d x

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

Solution steps

\int cos (x) (1 + 7 sin^{2} (x)) d x

Apply u-substitution

: \int 1 + 7 u^{2} d u

= \int 1 + 7 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 7 u^{2} d u

\int 1 d u = u

\int 7 u^{2} d u = \frac{7 u ^{3}}{3}

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

Substitute back

u = sin (x)

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

Add a constant to the solution

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

ln (2 x)

n = 0 \sum \infty \frac{1}{n ^{4} + 6}

f (x) = x, at x = 5

f^{^{''}} (t) - f (t) = - 8 t, f (3) = - 8

f (x) = \frac{2 x - 1}{4 x + 8}

What is the integral of cos(x)(1+7sin^2(x)) ?
The integral of cos(x)(1+7sin^2(x)) is sin(x)+(7sin^3(x))/3+C