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

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

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integral of (x^3sin(x))

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

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

Solution steps

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

Apply Integration By Parts

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

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

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

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

Simplify

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

Add a constant to the solution

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

\int \frac{e ^{x}}{e ^{- 3 x - 1}} d x

\int \frac{x ^{5}}{3 1 + x ^{3}} d x

\int (tan (2 x) cot (2 x))^{2} d x

\int x^{3} 3^{x^{4}} d x

\int - 5 y d y

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