What is the integral of 9x^3sin(x^2) ?

The integral of 9x^3sin(x^2) is 9/2 (-x^2cos(x^2)+sin(x^2))+C

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integral of 9x^3sin(x^2)

\int 9 x^{3} sin (x^{2}) d x

\frac{9}{2} (- x^{2} cos (x^{2}) + sin (x^{2})) + C

Solution steps

\int 9 x^{3} sin (x^{2}) d x

Take the constant out:

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

= 9 \cdot \int x^{3} sin (x^{2}) d x

Apply u-substitution

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

= 9 \cdot \int \frac{u sin ( u )}{2} d u

Take the constant out:

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

= 9 \cdot \frac{1}{2} \cdot \int u sin (u) d u

Apply Integration By Parts

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

= 9 \cdot \frac{1}{2} (- u cos (u) - \int - cos (u) d u)

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

= 9 \cdot \frac{1}{2} (- u cos (u) - (- sin (u)))

Substitute back

u = x^{2}

= 9 \cdot \frac{1}{2} (- x^{2} cos (x^{2}) - (- sin (x^{2})))

Simplify

9 \cdot \frac{1}{2} (- x^{2} cos (x^{2}) - (- sin (x^{2}))) : \frac{9}{2} (- x^{2} cos (x^{2}) + sin (x^{2}))

= \frac{9}{2} (- x^{2} cos (x^{2}) + sin (x^{2}))

Add a constant to the solution

= \frac{9}{2} (- x^{2} cos (x^{2}) + sin (x^{2})) + C

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

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f (x) = (x^{3} + x) (x^{2} + 2)

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

\int \frac{1}{( x - 3 ) 4 x ^{2} - 24 x + 20} d x

What is the integral of 9x^3sin(x^2) ?
The integral of 9x^3sin(x^2) is 9/2 (-x^2cos(x^2)+sin(x^2))+C