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Evaluate by method of substitution: ∫(6x+3 )/(x^2+x−5)^7dx

User Bjh Hans
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Final answer:

The given integral can be solved using the method of substitution. Let u = x^2 + x - 5, then du = (2x + 1) dx. Therefore, the integral is 3 ∫u^-7 du, which can be solved using the power rule for integrals.

Step-by-step explanation:

The integral ∫(6x+3 )/(x^2+x−5)^7dx can be evaluated using the method of substitution. In this case, we would let u = x^2 + x - 5. Then, du = (2x + 1) dx. But in the integral, we have (6x + 3) dx not (2x+1)dx. However, we notice that 6x + 3 = 3*2x + 3. This can be rewritten as 3(2x + 1). Hence, the integral can be rewritten as 3 ∫u^-7 du. This is now an easier integral to compute using the power rule for integrals: ∫u^n du = (u^n+1/(n+1)) + C, where C is the constant of integration.

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