# How do you use substitution to integrate # (2x+3)/(x+7)^3 #?

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To integrate (\frac{{2x + 3}}{{(x + 7)^3}}) using substitution, let (u = x + 7). Then, (du = dx).

Rewrite the integral with respect to (u): [\int \frac{{2(u - 7) + 3}}{{u^3}} du]

Simplify the integrand: [\int \frac{{2u - 11}}{{u^3}} du]

Split the fraction into two separate fractions: [\int \frac{{2u}}{{u^3}} du - \int \frac{{11}}{{u^3}} du]

Integrate each term separately: [\int \frac{2}{{u^2}} du - 11 \int \frac{1}{{u^3}} du]

Using power rule for integration: [= -\frac{2}{u} + \frac{11}{2u^2} + C]

Replace (u) with (x + 7): [= -\frac{2}{{x + 7}} + \frac{{11}}{{2(x + 7)^2}} + C]

Therefore, the integral of (\frac{{2x + 3}}{{(x + 7)^3}}) is (-\frac{2}{{x + 7}} + \frac{{11}}{{2(x + 7)^2}} + C), where (C) is the constant of integration.

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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.

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.

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.

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