Question

Find the value of \(\int \frac{1}{4x^2+4x+5} dx\).

(a) ^1⁄8 sin^(-1)⁡(x + ^1⁄2)

(b) ^1⁄4 tan^(-1)⁡(x + ^1⁄2)

(c) ^1⁄8 sec^(-1)⁡(x + ^1⁄2)

(d) ^1⁄4 cos^(-1)⁡(x + ^1⁄2)

This question was posed to me in an interview.

Query is from Improper Integrals in division Integral Calculus of Engineering Mathematics

Answer 1

Right option is (b) ^1⁄4 tan^(-1)⁡(x + ^1⁄2)

Explanation: Add constant automatically

Given, \(\int \frac{1}{4x^2+4x+5} dx\)

=\(\int \frac{1}{4 (x^2+x+\frac{5}{4}+\frac{1}{4}+\frac{1}{4})} dx =\int \frac{1}{4[(x+\frac{1}{2})^2+1^2])}dx=\frac{1}{4} tan^{-1}(x+\frac{1}{2})\)