What is the integral of cos(x)(6+7sin^2(x)) ?

The integral of cos(x)(6+7sin^2(x)) is 6sin(x)+(7sin^3(x))/3+C

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integral of cos(x)(6+7sin^2(x))

\int cos (x) (6 + 7 sin^{2} (x)) d x

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

Solution steps

\int cos (x) (6 + 7 sin^{2} (x)) d x

Apply u-substitution

: \int 6 + 7 u^{2} d u

= \int 6 + 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 6 d u + \int 7 u^{2} d u

\int 6 d u = 6 u

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

= 6 u + \frac{7 u ^{3}}{3}

Substitute back

u = sin (x)

= 6 sin (x) + \frac{7 sin ^{3} ( x )}{3}

Add a constant to the solution

= 6 sin (x) + \frac{7 sin ^{3} ( x )}{3} + C

\frac{\partial}{\partial y} (x sin (y))

\frac{d}{d x} (\frac{1}{( 1 + x ^{2} ) ^{2}})

\frac{d y}{d x}, x y = cot (x y)

\frac{\partial}{\partial x} (\frac{z}{x} ln (y) + ln (x))

\int \frac{x}{2 + x} d x

What is the integral of cos(x)(6+7sin^2(x)) ?
The integral of cos(x)(6+7sin^2(x)) is 6sin(x)+(7sin^3(x))/3+C