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

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

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integral of xsin^3(x^2)

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

\frac{1}{2} (- cos (x^{2}) + \frac{cos ^{3} ( x ^{2} )}{3}) + C

Solution steps

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

Apply u-substitution

: \int \frac{sin ^{3} ( u )}{2} d u

= \int \frac{sin ^{3} ( 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 sin^{3} (u) d u

Rewrite using trig identities

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

Apply u-substitution

: \int - 1 + v^{2} d v

= \frac{1}{2} \cdot \int - 1 + v^{2} d v

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{2} (- \int 1 d v + \int v^{2} d v)

\int 1 d v = v

\int v^{2} d v = \frac{v ^{3}}{3}

= \frac{1}{2} (- v + \frac{v ^{3}}{3})

Substitute back

= \frac{1}{2} (- cos (x^{2}) + \frac{cos ^{3} ( x ^{2} )}{3})

Add a constant to the solution

= \frac{1}{2} (- cos (x^{2}) + \frac{cos ^{3} ( x ^{2} )}{3}) + C

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