What is the integral of x(2x+3)^7 ?

The integral of x(2x+3)^7 is (128x^9)/9+168x^8+864x^7+2520x^6+4536x^5+5103x^4+3402x^3+(2187x^2)/2+C

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integral of x(2x+3)^7

\int x (2 x + 3)^{7} d x

\frac{128 x ^{9}}{9} + 168 x^{8} + 864 x^{7} + 2520 x^{6} + 4536 x^{5} + 5103 x^{4} + 3402 x^{3} + \frac{2187 x ^{2}}{2} + C

Solution steps

\int x (2 x + 3)^{7} d x

Expand

x (2 x + 3)^{7} : 128 x^{8} + 1344 x^{7} + 6048 x^{6} + 15120 x^{5} + 22680 x^{4} + 20412 x^{3} + 10206 x^{2} + 2187 x

= \int 128 x^{8} + 1344 x^{7} + 6048 x^{6} + 15120 x^{5} + 22680 x^{4} + 20412 x^{3} + 10206 x^{2} + 2187 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 128 x^{8} d x + \int 1344 x^{7} d x + \int 6048 x^{6} d x + \int 15120 x^{5} d x + \int 22680 x^{4} d x + \int 20412 x^{3} d x + \int 10206 x^{2} d x + \int 2187 x d x

\int 128 x^{8} d x = \frac{128 x ^{9}}{9}

\int 1344 x^{7} d x = 168 x^{8}

\int 6048 x^{6} d x = 864 x^{7}

\int 15120 x^{5} d x = 2520 x^{6}

\int 22680 x^{4} d x = 4536 x^{5}

\int 20412 x^{3} d x = 5103 x^{4}

\int 10206 x^{2} d x = 3402 x^{3}

\int 2187 x d x = \frac{2187 x ^{2}}{2}

= \frac{128 x ^{9}}{9} + 168 x^{8} + 864 x^{7} + 2520 x^{6} + 4536 x^{5} + 5103 x^{4} + 3402 x^{3} + \frac{2187 x ^{2}}{2}

Add a constant to the solution

= \frac{128 x ^{9}}{9} + 168 x^{8} + 864 x^{7} + 2520 x^{6} + 4536 x^{5} + 5103 x^{4} + 3402 x^{3} + \frac{2187 x ^{2}}{2} + C

\int a x^{7} d x

\int \frac{cot ( t ) csc ( t )}{csc ^{7} ( t )} d t

\int - tan (2 x + 1) ln (cos (2 x + 1)) d x

\int \frac{2 x - 7}{x ^{2} - 12 x + 32} d x

\int \frac{e ^{3 x}}{( e ^{x} + 1 ) ^{2}} d x

What is the integral of x(2x+3)^7 ?
The integral of x(2x+3)^7 is (128x^9)/9+168x^8+864x^7+2520x^6+4536x^5+5103x^4+3402x^3+(2187x^2)/2+C