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

The integral from 0 to 1 of (3-2x)^{-2} is 1/3

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integral from 0 to 1 of (3-2x)^{-2}

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

\frac{1}{3}

Decimal Notation

0.33333 \dots

Solution steps

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

Apply exponent rule:

a^{- b} = \frac{1}{a ^{b}}

= \int_{0}^{1} \frac{1}{( 3 - 2 x ) ^{2}} d x

Apply u-substitution

: \int_{3}^{1} - \frac{1}{2 u ^{2}} d u

= \int_{3}^{1} - \frac{1}{2 u ^{2}} d u

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

= - \int_{1}^{3} - \frac{1}{2 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_{1}^{3} \frac{1}{u ^{2}} d u)

Apply the Power Rule

: [- \frac{1}{u}]_{1}^{3}

= - (- \frac{1}{2} [- \frac{1}{u}]_{1}^{3})

Simplify

= \frac{1}{2} [- \frac{1}{u}]_{1}^{3}

Compute the boundaries:

\frac{2}{3}

= \frac{1}{2} \cdot \frac{2}{3}

Simplify

= \frac{1}{3}

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What is the integral from 0 to 1 of (3-2x)^{-2} ?
The integral from 0 to 1 of (3-2x)^{-2} is 1/3