What is the integral of sin(x)(2x^2+cos^3(x)) ?

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

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integral of sin(x)(2x^2+cos^3(x))

\int sin (x) (2 x^{2} + cos^{3} (x)) d x

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

Solution steps

\int sin (x) (2 x^{2} + cos^{3} (x)) d x

Expand

sin (x) (2 x^{2} + cos^{3} (x)) : 2 x^{2} sin (x) + cos^{3} (x) sin (x)

= \int 2 x^{2} sin (x) + cos^{3} (x) sin (x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 2 x^{2} sin (x) d x + \int cos^{3} (x) sin (x) d x

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

\int cos^{3} (x) sin (x) d x = - \frac{cos ^{4} ( x )}{4}

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

Add a constant to the solution

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

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