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How do you integrate #int (2x+5)(x^2+5x)^7dx# using substitution?

Answer 1

Emma Aldridge

#int(2x+5)(x^2+5x)^7dx=1/8(x^2+5x)^8+"c"#

We want to find #int(2x+5)(x^2+5x)^7dx=int(x^2+5x)^7(2x+5)dx#.

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

#intu^7du=1/8u^8+"c"=1/8(x^2+5x)^8+"c"#

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Answer 2

Christopher Agresta

To integrate ( \int (2x+5)(x^2+5x)^7 , dx ) using substitution, let ( u = x^2 + 5x ). Then, ( du = (2x + 5) , dx ). Rewrite the integral in terms of ( u ), which gives ( \int u^7 , du ). Integrating ( u^7 ) with respect to ( u ) gives ( \frac{u^8}{8} + C ). Substitute ( x^2 + 5x ) back in for ( u ) to get ( \frac{(x^2+5x)^8}{8} + C ), where ( C ) is the constant of integration.

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Answer 3

Christopher Agresta

To integrate (\int (2x+5)(x^2+5x)^7 , dx) using substitution, let (u = x^2 + 5x). Then, (du = (2x + 5) , dx). After substituting, the integral becomes (\int u^7 , du). Integrate (u^7) with respect to (u), and then substitute back for (x) to find the final result.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.