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

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

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

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

- x^{2} cos (x) - 3 x cos (x) + 2 x sin (x) + 3 sin (x) + 2 cos (x) + C

Solution steps

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

Apply Integration By Parts

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

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

\int - cos (x) (2 x + 3) d x = - 2 x sin (x) - 3 sin (x) - 2 cos (x)

= - cos (x) (x^{2} + 3 x) - (- 2 x sin (x) - 3 sin (x) - 2 cos (x))

Simplify

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

= - x^{2} cos (x) - 3 x cos (x) + 2 x sin (x) + 3 sin (x) + 2 cos (x)

Add a constant to the solution

= - x^{2} cos (x) - 3 x cos (x) + 2 x sin (x) + 3 sin (x) + 2 cos (x) + C

\int \frac{1}{x ( 1 + x ^{2} ) ^{\frac{3}{2}}} d x

\int \frac{1 - x}{- x ^{2}} d x

\int \frac{1}{2 y + 5} d y

\int \frac{9}{1 - x ^{2}} d x

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