What is the integral from 0 to a of x^3f(x^2) ?

The integral from 0 to a of x^3f(x^2) is (a^6f)/6

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integral from 0 to a of x^3f(x^2)

\int_{0}^{a} x^{3} f (x^{2}) d x

\frac{a ^{6} f}{6}

Solution steps

\int_{0}^{a} x^{3} f x^{2} d x

Take the constant out:

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

= f \cdot \int_{0}^{a} x^{2} x^{3} d x

Simplify

x^{2} x^{3} : x^{5}

= f \cdot \int_{0}^{a} x^{5} d x

Apply the Power Rule

: [\frac{x ^{6}}{6}]_{0}^{a}

= f [\frac{x ^{6}}{6}]_{0}^{a}

Compute the boundaries:

\frac{a ^{6}}{6}

= f \frac{a ^{6}}{6}

Simplify

= \frac{a ^{6} f}{6}

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What is the integral from 0 to a of x^3f(x^2) ?
The integral from 0 to a of x^3f(x^2) is (a^6f)/6