What is the integral from 0 to 1 of x/((x+15)^2) ?

The integral from 0 to 1 of x/((x+15)^2) is 4ln(2)-ln(15)-1/16

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integral from 0 to 1 of x/((x+15)^2)

\int_{0}^{1} \frac{x}{( x + 15 ) ^{2}} d x

4 ln (2) - ln (15) - \frac{1}{16}

Decimal

0.00203 \dots

Solution steps

\int_{0}^{1} \frac{x}{( x + 15 ) ^{2}} d x

Apply u-substitution

: \int_{15}^{16} \frac{u - 15}{u ^{2}} d u

= \int_{15}^{16} \frac{u - 15}{u ^{2}} d u

Expand

\frac{u - 15}{u ^{2}} : \frac{1}{u} - \frac{15}{u ^{2}}

= \int_{15}^{16} \frac{1}{u} - \frac{15}{u ^{2}} d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{15}^{16} \frac{1}{u} d u - \int_{15}^{16} \frac{15}{u ^{2}} d u

\int_{15}^{16} \frac{1}{u} d u = 4 ln (2) - ln (15)

\int_{15}^{16} \frac{15}{u ^{2}} d u = \frac{1}{16}

= 4 ln (2) - ln (15) - \frac{1}{16}

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What is the integral from 0 to 1 of x/((x+15)^2) ?
The integral from 0 to 1 of x/((x+15)^2) is 4ln(2)-ln(15)-1/16