What is the integral of x^3sin(x^2+1) ?

The integral of x^3sin(x^2+1) is 1/2 (-x^2cos(x^2+1)+sin(x^2+1))+C

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integral of x^3sin(x^2+1)

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

\frac{1}{2} (- x^{2} cos (x^{2} + 1) + sin (x^{2} + 1)) + C

Solution steps

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

Apply u-substitution

: \int \frac{( u - 1 ) sin ( u )}{2} d u

= \int \frac{( u - 1 ) sin ( u )}{2} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int (u - 1) sin (u) d u

Apply Integration By Parts

: - cos (u) (u - 1) - \int - cos (u) d u

= \frac{1}{2} (- cos (u) (u - 1) - \int - cos (u) d u)

\int - cos (u) d u = - sin (u)

= \frac{1}{2} (- cos (u) (u - 1) - (- sin (u)))

Substitute back

u = x^{2} + 1

= \frac{1}{2} (- cos (x^{2} + 1) (x^{2} + 1 - 1) - (- sin (x^{2} + 1)))

Simplify

\frac{1}{2} (- cos (x^{2} + 1) (x^{2} + 1 - 1) - (- sin (x^{2} + 1))) : \frac{1}{2} (- x^{2} cos (x^{2} + 1) + sin (x^{2} + 1))

= \frac{1}{2} (- x^{2} cos (x^{2} + 1) + sin (x^{2} + 1))

Add a constant to the solution

= \frac{1}{2} (- x^{2} cos (x^{2} + 1) + sin (x^{2} + 1)) + C

\int \frac{5 x ^{3} - 8}{x ^{3}} d x

\int ln (2 x^{2} + 5) d x

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

\int e^{x} arcsin (1 - e^{2 x}) d x

What is the integral of x^3sin(x^2+1) ?
The integral of x^3sin(x^2+1) is 1/2 (-x^2cos(x^2+1)+sin(x^2+1))+C