Question

Evaluate the following integrals

(a)

\int_{0}^{\infty }\frac{x^{2}+2}{(x^{2}+1)(x^{2}+4)}dx

(b)\int_{-\infty }^{\infty }\frac{sin^{2}2x}{1+x^{2}}dx

22 Mar 2023
Answer :
Word Count : 613

(a) To evaluate the integral, we will use partial fraction decomposition. We will rewrite the integrand as:

\frac{x^2 + 2}{(x^2 + 1)(x^2 + 4)} = \frac{A}{x^2 + 1} + \frac{B}{x^2 + 4}

where A and B are constants. To find the values of A and B, we need to multiply both sides by (x^2 + 1)(x^2 + 4) and solve for A and B:

x^2 + 2 = A(x^2 + 4) + B(x^2 + 1)

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