What is the integral of cos(x)(4+7sin^2(x)) ?

The integral of cos(x)(4+7sin^2(x)) is 4sin(x)+(7sin^3(x))/3+C

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integral of cos(x)(4+7sin^2(x))

\int cos (x) (4 + 7 sin^{2} (x)) d x

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

Solution steps

\int cos (x) (4 + 7 sin^{2} (x)) d x

Apply u-substitution

: \int 4 + 7 u^{2} d u

= \int 4 + 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 4 d u + \int 7 u^{2} d u

\int 4 d u = 4 u

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

= 4 u + \frac{7 u ^{3}}{3}

Substitute back

u = sin (x)

= 4 sin (x) + \frac{7 sin ^{3} ( x )}{3}

Add a constant to the solution

= 4 sin (x) + \frac{7 sin ^{3} ( x )}{3} + C

\frac{\partial}{\partial y} (3 x y^{2} z^{2})

f^{^{'}} (t) = \frac{6}{t ^{6}} - \frac{3}{t ^{3}} + \frac{2}{t}

\frac{d}{d x} (3 sin (x))

f (y) = sin (y)

x \to 1 lim (e^{2 x})

What is the integral of cos(x)(4+7sin^2(x)) ?
The integral of cos(x)(4+7sin^2(x)) is 4sin(x)+(7sin^3(x))/3+C