/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 43 Evaluate the following integrals... [FREE SOLUTION] | 91影视

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Chapter 7: Problem 43

Evaluate the following integrals. $$\int \frac{x^{2}+x+2}{(x+1)\left(x^{2}+1\right)} d x$$

Short Answer

Expert verified
Question: Evaluate the integral of the following rational function: $$\int \frac{x^{2}+x+2}{(x+1)\left(x^{2}+1\right)} d x$$ Answer: $$\int \frac{x^{2}+x+2}{(x+1)\left(x^{2}+1\right)} d x = \ln |x+1| + \arctan x + C$$

Step by step solution

01

Perform partial fraction decomposition

We need to decompose the given rational function into simpler fractions. The given function is: $$\frac{x^{2}+x+2}{(x+1)\left(x^{2}+1\right)}$$ We need to find the constants A, B, and C such that: $$\frac{x^{2}+x+2}{(x+1)\left(x^{2}+1\right)} = \frac{A}{x+1} + \frac{Bx + C}{x^2 + 1}$$ Clearing the denominators, we get: $$x^2 + x + 2 = A(x^2 + 1) + (Bx + C)(x + 1)$$
02

Solve for A, B, and C

To find A, B, and C, we can substitute different values of x: When x = -1: $$A = \frac{-1+0+2}{0+1} = 1$$ Now, we will expand the polynomial equation: $$x^2 +x+2 = Ax^2 + A + (Bx^2 + Cx + Bx + C)$$ $$x^2 + x + 2 = (A + B)x^2 + (C + B)x + (A + C)$$ Comparing coefficients of like terms, we have: 1) \(A + B = 1\) 2) \(C + B = 1\) 3) \(A + C = 2\) Since A = 1 from our previous substitution, we know B = 0 and C = 1. Plugging A, B, and C back into the simplified fractions, we have: $$\int \frac{x^{2}+x+2}{(x+1)\left(x^{2}+1\right)} d x = \int \left(\frac{1}{x+1} + \frac{0x + 1}{x^2 + 1}\right) dx$$
03

Integrate the decomposed terms

Now, we integrate the simplified terms: $$ = \int \frac{1}{x+1} dx + \int \frac{1}{x^2 + 1} dx$$ The integrals can be found by recognizing the antiderivative forms: For the first term, \(\int \frac{1}{x+1} dx = \ln |x+1| + C_1\) For the second term, \(\int \frac{1}{x^2 + 1} dx = \arctan x + C_2\)
04

Combine the antiderivatives

Now, we combine the antiderivative terms and simplify to write our final answer: $$ = \ln |x+1| + \arctan x + C$$ Thus, the solution to the given integral is: $$\int \frac{x^{2}+x+2}{(x+1)\left(x^{2}+1\right)} d x = \ln |x+1| + \arctan x + C$$

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