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

The integral of xsin^3(x^2)cos(x^2) is 1/8 sin^4(x^2)+C

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integral of xsin^3(x^2)cos(x^2)

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

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

Solution steps

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

Apply u-substitution

Take the constant out:

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

Apply the Power Rule

: \frac{3}{4} u^{\frac{4}{3}}

= \frac{1}{6} \cdot \frac{3}{4} u^{\frac{4}{3}}

Substitute back

u = sin^{3} (x^{2})

= \frac{1}{6} \cdot \frac{3}{4} (sin^{3} (x^{2}))^{\frac{4}{3}}

Simplify

\frac{1}{6} \cdot \frac{3}{4} (sin^{3} (x^{2}))^{\frac{4}{3}} : \frac{1}{8} sin^{4} (x^{2})

= \frac{1}{8} sin^{4} (x^{2})

Add a constant to the solution

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

\frac{d y}{d x} = x, y (11) = 5

y = 5 sec (x), at x = \frac{π}{3}

\frac{d}{d x} (- 5 e^{- 4 x})

\int_{0}^{2} (x + 1) d x

y = 1 + \frac{2}{x ^{4}}

What is the integral of xsin^3(x^2)cos(x^2) ?
The integral of xsin^3(x^2)cos(x^2) is 1/8 sin^4(x^2)+C