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

The integral from 0 to 1 of 5/((2x+1)^2) is 5/3

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

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

\frac{5}{3}

Decimal

1.66666 \dots

Solution steps

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

Take the constant out:

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

= 5 \cdot \int_{0}^{1} \frac{1}{( 2 x + 1 ) ^{2}} d x

Apply u-substitution

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

= 5 \cdot \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

= 5 \cdot \frac{1}{2} \cdot \int_{1}^{3} \frac{1}{u ^{2}} d u

Apply the Power Rule

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

= 5 \cdot \frac{1}{2} [- \frac{1}{u}]_{1}^{3}

Simplify

5 \cdot \frac{1}{2} [- \frac{1}{u}]_{1}^{3} : \frac{5}{2} [- \frac{1}{u}]_{1}^{3}

= \frac{5}{2} [- \frac{1}{u}]_{1}^{3}

Compute the boundaries:

\frac{2}{3}

= \frac{5}{2} \cdot \frac{2}{3}

Simplify

= \frac{5}{3}

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What is the integral from 0 to 1 of 5/((2x+1)^2) ?
The integral from 0 to 1 of 5/((2x+1)^2) is 5/3