What is the integral from 0 to 1 of 1/((5-4x)^2) ?

The integral from 0 to 1 of 1/((5-4x)^2) is 1/5

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integral from 0 to 1 of 1/((5-4x)^2)

\int_{0}^{1} \frac{1}{( 5 - 4 x ) ^{2}} d x

\frac{1}{5}

Decimal

0.2

Solution steps

\int_{0}^{1} \frac{1}{( 5 - 4 x ) ^{2}} d x

Apply u-substitution

: \int_{5}^{1} - \frac{1}{4 u ^{2}} d u

= \int_{5}^{1} - \frac{1}{4 u ^{2}} d u

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

= - \int_{1}^{5} - \frac{1}{4 u ^{2}} d u

Take the constant out:

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

= - (- \frac{1}{4} \cdot \int_{1}^{5} \frac{1}{u ^{2}} d u)

Apply the Power Rule

: [- \frac{1}{u}]_{1}^{5}

= - (- \frac{1}{4} [- \frac{1}{u}]_{1}^{5})

Simplify

= \frac{1}{4} [- \frac{1}{u}]_{1}^{5}

Compute the boundaries:

\frac{4}{5}

= \frac{1}{4} \cdot \frac{4}{5}

Simplify

= \frac{1}{5}

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What is the integral from 0 to 1 of 1/((5-4x)^2) ?
The integral from 0 to 1 of 1/((5-4x)^2) is 1/5