What is the integral of x^2ln(2x+5) ?

The integral of x^2ln(2x+5) is 1/3 x^3ln(2x+5)-1/72 (8x^3-30x^2+150x-375ln|2x+5|)+C

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integral of x^2ln(2x+5)

\int x^{2} ln (2 x + 5) d x

\frac{1}{3} x^{3} ln (2 x + 5) - \frac{1}{72} (8 x^{3} - 30 x^{2} + 150 x - 375 ln ∣ 2 x + 5 ∣) + C

Solution steps

\int x^{2} ln (2 x + 5) d x

Apply Integration By Parts

: \frac{1}{3} x^{3} ln (2 x + 5) - \int \frac{2 x ^{3}}{3 ( 2 x + 5 )} d x

= \frac{1}{3} x^{3} ln (2 x + 5) - \int \frac{2 x ^{3}}{3 ( 2 x + 5 )} d x

\int \frac{2 x ^{3}}{3 ( 2 x + 5 )} d x = \frac{1}{72} (8 x^{3} - 30 x^{2} + 150 x - 375 ln ∣ 2 x + 5 ∣)

= \frac{1}{3} x^{3} ln (2 x + 5) - \frac{1}{72} (8 x^{3} - 30 x^{2} + 150 x - 375 ln ∣ 2 x + 5 ∣)

Add a constant to the solution

= \frac{1}{3} x^{3} ln (2 x + 5) - \frac{1}{72} (8 x^{3} - 30 x^{2} + 150 x - 375 ln ∣ 2 x + 5 ∣) + C

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What is the integral of x^2ln(2x+5) ?
The integral of x^2ln(2x+5) is 1/3 x^3ln(2x+5)-1/72 (8x^3-30x^2+150x-375ln|2x+5|)+C