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

The integral of x(3x+2)^5 is (243x^7)/7+135x^6+216x^5+180x^4+80x^3+16x^2+C

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integral of x(3x+2)^5

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

\frac{243 x ^{7}}{7} + 135 x^{6} + 216 x^{5} + 180 x^{4} + 80 x^{3} + 16 x^{2} + C

Solution steps

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

Expand

x (3 x + 2)^{5} : 243 x^{6} + 810 x^{5} + 1080 x^{4} + 720 x^{3} + 240 x^{2} + 32 x

= \int 243 x^{6} + 810 x^{5} + 1080 x^{4} + 720 x^{3} + 240 x^{2} + 32 x 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 243 x^{6} d x + \int 810 x^{5} d x + \int 1080 x^{4} d x + \int 720 x^{3} d x + \int 240 x^{2} d x + \int 32 x d x

\int 243 x^{6} d x = \frac{243 x ^{7}}{7}

\int 810 x^{5} d x = 135 x^{6}

\int 1080 x^{4} d x = 216 x^{5}

\int 720 x^{3} d x = 180 x^{4}

\int 240 x^{2} d x = 80 x^{3}

\int 32 x d x = 16 x^{2}

= \frac{243 x ^{7}}{7} + 135 x^{6} + 216 x^{5} + 180 x^{4} + 80 x^{3} + 16 x^{2}

Add a constant to the solution

= \frac{243 x ^{7}}{7} + 135 x^{6} + 216 x^{5} + 180 x^{4} + 80 x^{3} + 16 x^{2} + C

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What is the integral of x(3x+2)^5 ?
The integral of x(3x+2)^5 is (243x^7)/7+135x^6+216x^5+180x^4+80x^3+16x^2+C