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

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

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

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

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

Solution steps

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

Apply Integration By Parts

: (2 + 4 sin^{2} (x)) sin (x) - \int 4 sin (2 x) sin (x) d x

= (2 + 4 sin^{2} (x)) sin (x) - \int 4 sin (2 x) sin (x) d x

\int 4 sin (2 x) sin (x) d x = \frac{8}{3} sin^{3} (x)

= (2 + 4 sin^{2} (x)) sin (x) - \frac{8}{3} sin^{3} (x)

Simplify

(2 + 4 sin^{2} (x)) sin (x) - \frac{8}{3} sin^{3} (x) : 2 sin (x) + \frac{4}{3} sin^{3} (x)

= 2 sin (x) + \frac{4}{3} sin^{3} (x)

Add a constant to the solution

= 2 sin (x) + \frac{4}{3} sin^{3} (x) + C

f (x) = \frac{2 x}{1 + x ^{2}}, at x = (4)

x y^{^{'}} - 2 y = x^{4}, y (1) = 0

\frac{d}{d x} ((4 x - 8) e^{4 x})

x^{5 π}

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