int(sqrt(x+1)+2)/((x+1)^(2)-sqrt(x+1))dx

Question

Bytelearn · Accepted Answer

Factorize Denominator: Simplify the integrand by factoring the denominator. The denominator is a difference of squares: (x+1)^(2) - sqrt(x+1) can be factored as ((x+1) - sqrt(x+1))((x+1) + sqrt(x+1)).Rewrite Numerator: Notice that the numerator can be rewritten to match the factors in the denominator. The numerator is sqrt(x+1) + 2, which can be written as (sqrt(x+1) + 1) + 1 to match the factors (x+1) - sqrt(x+1) and (x+1) + sqrt(x+1) in the denominator.Split into Partial Fractions: Split the integral into two parts using partial fractions. We can write the integrand as A/((x+1) - sqrt(x+1)) + B/((x+1) + sqrt(x+1)), where A and B are constants to be determined.Solve for Constants: Solve for A and B by setting up equations. We have (sqrt(x+1) + 2) = A((x+1) + sqrt(x+1)) + B((x+1) - sqrt(x+1)). We need to find the values of A and B that satisfy this equation.Find A and B: Plug in convenient values for x to solve for A and B. Let x = -1 to solve for B. We get 2 = 2B, so B = 1. Let x be such that sqrt(x+1) = 1 (x = 0) to solve for A. We get 3 = 2A + B, so 3 = 2A + 1, which gives A = 1.Rewrite with A and B: Rewrite the integral with the found values of A and B. The integral becomes int 1/((x+1) - sqrt(x+1)) dx + int 1/((x+1) + sqrt(x+1)) dx.Integrate Separately: Integrate each term separately. For the first integral, let u = (x+1) - sqrt(x+1), then du = (1 - (1/(2sqrt(x+1)))) dx. For the second integral, let v = (x+1) + sqrt(x+1), then dv = (1 + (1/(2sqrt(x+1)))) dx.Perform Substitution: Perform the substitution for both integrals. The first integral becomes int 1/u du, and the second integral becomes int 1/v dv.Integrate with Respect: Integrate with respect to u and v. The first integral is ln|u| + C, and the second integral is ln|v| + D, where C and D are constants of integration.Substitute Back: Substitute back the original variables. We have ln|(x+1) - sqrt(x+1)| + ln|(x+1) + sqrt(x+1)| + E, where E is the combined constant of integration.Combine Logarithms: Combine the logarithms. The final answer is ln|((x+1) - sqrt(x+1))((x+1) + sqrt(x+1))| + E.Simplify Inside Log: Simplify the expression inside the logarithm. The expression inside the logarithm simplifies to (x+1)^2 - (sqrt(x+1))^2, which is x+1.Write Final Answer: Write the final simplified answer. The final answer is ln|x+1| + E.

$\int\frac{\sqrt{x+1}+2}{(x+1)^{2}-\sqrt{x+1}}dx$

Full solution

More problems from Evaluate definite integrals using the chain rule

Related topics

∫x+1+2(x+1)2−x+1dx\int\frac{\sqrt{x+1}+2}{(x+1)^{2}-\sqrt{x+1}}dx∫(x+1)2−x+1​x+1​+2​dx

$\int\frac{\sqrt{x+1}+2}{(x+1)^{2}-\sqrt{x+1}}dx$