What is the integral from 0 to 1 of x^2(3-10x^3)^4 ?

The integral from 0 to 1 of x^2(3-10x^3)^4 is 341/3

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integral from 0 to 1 of x^2(3-10x^3)^4

\int_{0}^{1} x^{2} (3 - 10 x^{3})^{4} d x

\frac{341}{3}

Decimal

113.66666 \dots

Solution steps

\int_{0}^{1} x^{2} (3 - 10 x^{3})^{4} d x

Apply u-substitution

: \int_{3}^{- 7} - \frac{u ^{4}}{30} d u

= \int_{3}^{- 7} - \frac{u ^{4}}{30} d u

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= - \int_{- 7}^{3} - \frac{u ^{4}}{30} d u

Take the constant out:

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

= - (- \frac{1}{30} \cdot \int_{- 7}^{3} u^{4} d u)

Apply the Power Rule

: [\frac{u ^{5}}{5}]_{- 7}^{3}

= - (- \frac{1}{30} [\frac{u ^{5}}{5}]_{- 7}^{3})

Simplify

= \frac{1}{30} [\frac{u ^{5}}{5}]_{- 7}^{3}

Compute the boundaries:

3410

= \frac{1}{30} \cdot 3410

Simplify

= \frac{341}{3}

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What is the integral from 0 to 1 of x^2(3-10x^3)^4 ?
The integral from 0 to 1 of x^2(3-10x^3)^4 is 341/3