QuestionJuly 14, 2025

Fill in the Blank 1 point int (x+1)/((x^2)+2x-3)^(2)dx=type your answer...

Fill in the Blank 1 point int (x+1)/((x^2)+2x-3)^(2)dx=type your answer...
Fill in the Blank 1 point int (x+1)/((x^2)+2x-3)^(2)dx=type your answer...

Solution
3.8(181 votes)

Answer

\frac{-1}{16(x-1)} + \frac{1}{16(x-1)^2} + \frac{1}{16(x+3)} + \frac{1}{16(x+3)^2} + C Explanation 1. Factor the denominator Factor x^2 + 2x - 3 as (x+3)(x-1). 2. Use partial fraction decomposition Express \frac{x+1}{(x+3)^2(x-1)^2} using partial fractions: A/(x+3) + B/(x+3)^2 + C/(x-1) + D/(x-1)^2. 3. Solve for coefficients Equate and solve for A, B, C, D by matching coefficients. 4. Integrate each term separately Integrate terms like \int \frac{A}{x+3} dx, \int \frac{B}{(x+3)^2} dx, etc., using basic integration formulas.

Explanation

1. Factor the denominator<br /> Factor $x^2 + 2x - 3$ as $(x+3)(x-1)$.<br />2. Use partial fraction decomposition<br /> Express $\frac{x+1}{(x+3)^2(x-1)^2}$ using partial fractions: $A/(x+3) + B/(x+3)^2 + C/(x-1) + D/(x-1)^2$.<br />3. Solve for coefficients<br /> Equate and solve for $A, B, C, D$ by matching coefficients.<br />4. Integrate each term separately<br /> Integrate terms like $\int \frac{A}{x+3} dx$, $\int \frac{B}{(x+3)^2} dx$, etc., using basic integration formulas.
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