How do you integrate $\int (2 x + 5) {(x^{2} + 5 x)}^{7} d x$ $int (2x+5)(x^2+5x)^7dx$ using substitution?

Question

5544 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2018-03-07T20:59:52

$\int (2 x + 5) {(x^{2} + 5 x)}^{7} d x = \frac{1}{8} {(x^{2} + 5 x)}^{8} + c$ $int(2x+5)(x^2+5x)^7dx=1/8(x^2+5x)^8+"c"$

We want to find $\int (2 x + 5) {(x^{2} + 5 x)}^{7} d x = \int {(x^{2} + 5 x)}^{7} (2 x + 5) d x$ $int(2x+5)(x^2+5x)^7dx=int(x^2+5x)^7(2x+5)dx$ .

Let $u = x^{2} + 5 x$ $u=x^2+5x$ and $d u = (2 x + 5) d x$ $du=(2x+5)dx$ and sub this into the integral to transform it to

$\int u^{7} d u = \frac{1}{8} u^{8} + c = \frac{1}{8} {(x^{2} + 5 x)}^{8} + c$ $intu^7du=1/8u^8+"c"=1/8(x^2+5x)^8+"c"$

How do you integrate ∫(2x+5)(x2+5x)7dxint (2x+5)(x^2+5x)^7dx using substitution?