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

The integral of x^3sin(2x^2) is

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

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

\frac{1}{2} sin^{2} (x^{2}) arcsin^{2} (sin (x^{2})) - \frac{1}{2} arcsin^{2} (sin (x^{2})) + \frac{1}{4} arcsin (sin (x^{2})) sin (2 arcsin (sin (x^{2}))) + \frac{1}{4} arcsin^{2} (sin (x^{2})) + \frac{1}{8} cos (2 arcsin (sin (x^{2}))) + C

Solution steps

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

Rewrite using trig identities

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

Take the constant out:

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

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

Apply u-substitution

= 2 \cdot \int \frac{arcsin ^{2} ( u ) u}{2} d u

Take the constant out:

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

= 2 \cdot \frac{1}{2} \cdot \int arcsin^{2} (u) u d u

Apply Integration By Parts

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

\int \frac{u ^{2} arcsin ( u )}{1 - u ^{2}} d u = \frac{1}{2} arcsin (u) (arcsin (u) - \frac{1}{2} sin (2 arcsin (u))) - \frac{1}{2} (\frac{arcsin ^{2} ( u )}{2} + \frac{1}{4} cos (2 arcsin (u)))

= 2 \cdot \frac{1}{2} (\frac{1}{2} u^{2} arcsin^{2} (u) - (\frac{1}{2} arcsin (u) (arcsin (u) - \frac{1}{2} sin (2 arcsin (u))) - \frac{1}{2} (\frac{arcsin ^{2} ( u )}{2} + \frac{1}{4} cos (2 arcsin (u)))))

Substitute back

u = sin (x^{2})

= 2 \cdot \frac{1}{2} (\frac{1}{2} sin^{2} (x^{2}) arcsin^{2} (sin (x^{2})) - (\frac{1}{2} arcsin (sin (x^{2})) (arcsin (sin (x^{2})) - \frac{1}{2} sin (2 arcsin (sin (x^{2})))) - \frac{1}{2} (\frac{arcsin ^{2} ( sin ( x ^{2} ) )}{2} + \frac{1}{4} cos (2 arcsin (sin (x^{2}))))))

Simplify

2 \cdot \frac{1}{2} (\frac{1}{2} sin^{2} (x^{2}) arcsin^{2} (sin (x^{2})) - (\frac{1}{2} arcsin (sin (x^{2})) (arcsin (sin (x^{2})) - \frac{1}{2} sin (2 arcsin (sin (x^{2})))) - \frac{1}{2} (\frac{arcsin ^{2} ( sin ( x ^{2} ) )}{2} + \frac{1}{4} cos (2 arcsin (sin (x^{2})))))) : \frac{1}{2} sin^{2} (x^{2}) arcsin^{2} (sin (x^{2})) - \frac{1}{2} arcsin^{2} (sin (x^{2})) + \frac{1}{4} arcsin (sin (x^{2})) sin (2 arcsin (sin (x^{2}))) + \frac{1}{4} arcsin^{2} (sin (x^{2})) + \frac{1}{8} cos (2 arcsin (sin (x^{2})))

= \frac{1}{2} sin^{2} (x^{2}) arcsin^{2} (sin (x^{2})) - \frac{1}{2} arcsin^{2} (sin (x^{2})) + \frac{1}{4} arcsin (sin (x^{2})) sin (2 arcsin (sin (x^{2}))) + \frac{1}{4} arcsin^{2} (sin (x^{2})) + \frac{1}{8} cos (2 arcsin (sin (x^{2})))

Add a constant to the solution

= \frac{1}{2} sin^{2} (x^{2}) arcsin^{2} (sin (x^{2})) - \frac{1}{2} arcsin^{2} (sin (x^{2})) + \frac{1}{4} arcsin (sin (x^{2})) sin (2 arcsin (sin (x^{2}))) + \frac{1}{4} arcsin^{2} (sin (x^{2})) + \frac{1}{8} cos (2 arcsin (sin (x^{2}))) + C