Evaluate the integral: \(∫_{-1}^1\frac {dx}{x^2+2x+5}\)
Solution and Explanation
\(∫_{-1}^1\frac {dx}{x^2+2x+5}\)= \(∫_{-1}^1\frac {dx}{(x^2+2x+1)+4}\) = \(∫_{-1}^1\frac {dx}{(x+1)^2+2^2}\)
\(Let\ x+1=t \implies dx=dt\)
\(When\ x=-1,t=0\ and\ when x=1,t=2\)
∴\(∫_{-1}^1\frac {dx}{(x+1)^2+2^2}\) = \(∫_0^2 \frac {dt}{t^2+2^2}\)
=\([\frac 12 tan^{-1}\frac t2]_0^2\)
=\(\frac 12 tan^{-1}1-\frac 12 tan^{-1}0\)
=\(\frac 12(\frac \pi4)\)
=\(\frac\pi8\)
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Concepts Used:
Integration by Partial Fractions
The number of formulas used to decompose the given improper rational functions is given below. By using the given expressions, we can quickly write the integrand as a sum of proper rational functions.
For examples,
