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

The integral from 0 to 5 of (x^2)/((5+2x)^2) is 5/4+(-15ln(3)+5)/(12)

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

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

\frac{5}{4} + \frac{- 15 ln ( 3 ) + 5}{12}

Decimal Notation

0.29340 \dots

Solution steps

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

Long division

\frac{x ^{2}}{( 5 + 2 x ) ^{2}} : \frac{1}{4} + \frac{- 20 x - 25}{4 ( 4 x ^{2} + 20 x + 25 )}

= \int_{0}^{5} \frac{1}{4} + \frac{- 20 x - 25}{4 ( 4 x ^{2} + 20 x + 25 )} d x

Apply the Sum Rule:

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

= \int_{0}^{5} \frac{1}{4} d x + \int_{0}^{5} \frac{- 20 x - 25}{4 ( 4 x ^{2} + 20 x + 25 )} d x

\int_{0}^{5} \frac{1}{4} d x = \frac{5}{4}

\int_{0}^{5} \frac{- 20 x - 25}{4 ( 4 x ^{2} + 20 x + 25 )} d x = \frac{- 15 ln ( 3 ) + 5}{12}

= \frac{5}{4} + \frac{- 15 ln ( 3 ) + 5}{12}

\int_{e}^{e^{10}} \frac{1}{x ln ( x )} d x

\int_{0}^{2} y 2 y - y^{2} d y

\int_{1}^{2} x^{2} (x - 2)^{10} d x

\int_{1}^{e} (17 + ln (x))^{2} d x

What is the integral from 0 to 5 of (x^2)/((5+2x)^2) ?
The integral from 0 to 5 of (x^2)/((5+2x)^2) is 5/4+(-15ln(3)+5)/(12)