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

The integral of x*(2x+3)^5 is (32x^7)/7+40x^6+144x^5+270x^4+270x^3+(243x^2)/2+C

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integral of x*(2x+3)^5

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

\frac{32 x ^{7}}{7} + 40 x^{6} + 144 x^{5} + 270 x^{4} + 270 x^{3} + \frac{243 x ^{2}}{2} + C

Solution steps

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

Expand

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

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

\int 32 x^{6} d x = \frac{32 x ^{7}}{7}

\int 240 x^{5} d x = 40 x^{6}

\int 720 x^{4} d x = 144 x^{5}

\int 1080 x^{3} d x = 270 x^{4}

\int 810 x^{2} d x = 270 x^{3}

\int 243 x d x = \frac{243 x ^{2}}{2}

= \frac{32 x ^{7}}{7} + 40 x^{6} + 144 x^{5} + 270 x^{4} + 270 x^{3} + \frac{243 x ^{2}}{2}

Add a constant to the solution

= \frac{32 x ^{7}}{7} + 40 x^{6} + 144 x^{5} + 270 x^{4} + 270 x^{3} + \frac{243 x ^{2}}{2} + C

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What is the integral of x*(2x+3)^5 ?
The integral of x*(2x+3)^5 is (32x^7)/7+40x^6+144x^5+270x^4+270x^3+(243x^2)/2+C