What is the integral from 0 to a of b/(a^2)(x-a)^2 ?

The integral from 0 to a of b/(a^2)(x-a)^2 is (ba)/3

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integral from 0 to a of b/(a^2)(x-a)^2

\int_{0}^{a} \frac{b}{a ^{2}} (x - a)^{2} d x

\frac{ba}{3}

Solution steps

\int_{0}^{a} \frac{b}{a ^{2}} (x - a)^{2} d x

Take the constant out:

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

= \frac{b}{a ^{2}} \cdot \int_{0}^{a} (x - a)^{2} d x

Apply u-substitution

: \int_{- a}^{0} u^{2} d u

= \frac{b}{a ^{2}} \cdot \int_{- a}^{0} u^{2} d u

Apply the Power Rule

: [\frac{u ^{3}}{3}]_{- a}^{0}

= \frac{b}{a ^{2}} [\frac{u ^{3}}{3}]_{- a}^{0}

Compute the boundaries:

\frac{a ^{3}}{3}

= \frac{b}{a ^{2}} \cdot \frac{a ^{3}}{3}

Simplify

= \frac{ba}{3}

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What is the integral from 0 to a of b/(a^2)(x-a)^2 ?
The integral from 0 to a of b/(a^2)(x-a)^2 is (ba)/3