What is the integral of x(3x+5)^2 ?

The integral of x(3x+5)^2 is 9/2 x^4-(9x^4)/4+10x^3+25/2 x^2+C

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integral of x(3x+5)^2

\int x (3 x + 5)^{2} d x

\frac{9}{2} x^{4} - \frac{9 x ^{4}}{4} + 10 x^{3} + \frac{25}{2} x^{2} + C

Solution steps

\int x (3 x + 5)^{2} d x

Apply Integration By Parts

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

\int 3 x^{2} (3 x + 5) d x = \frac{9 x ^{4}}{4} + 5 x^{3}

= \frac{1}{2} x^{2} (3 x + 5)^{2} - (\frac{9 x ^{4}}{4} + 5 x^{3})

Simplify

\frac{1}{2} x^{2} (3 x + 5)^{2} - (\frac{9 x ^{4}}{4} + 5 x^{3}) : \frac{9}{2} x^{4} - \frac{9 x ^{4}}{4} + 10 x^{3} + \frac{25}{2} x^{2}

= \frac{9}{2} x^{4} - \frac{9 x ^{4}}{4} + 10 x^{3} + \frac{25}{2} x^{2}

Add a constant to the solution

= \frac{9}{2} x^{4} - \frac{9 x ^{4}}{4} + 10 x^{3} + \frac{25}{2} x^{2} + C

What is the integral of x(3x+5)^2 ?
The integral of x(3x+5)^2 is 9/2 x^4-(9x^4)/4+10x^3+25/2 x^2+C