Final answer to the problem
$\frac{1}{2}\ln\left|2\right|+\frac{\pi }{8}-\frac{1}{4}\ln\left|2\right|$
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Step-by-step Solution
Learn how to solve integration by trigonometric substitution problems step by step online. Integrate the function 1/((x+1)(x^2+1)) from 0 to 1. Rewrite the fraction \frac{1}{\left(x+1\right)\left(x^2+1\right)} in 2 simpler fractions using partial fraction decomposition. Expand the integral \int_{0}^{1}\left(\frac{1}{2\left(x+1\right)}+\frac{-\frac{1}{2}x+\frac{1}{2}}{x^2+1}\right)dx into 2 integrals using the sum rule for integrals, to then solve each integral separately. The integral \int_{0}^{1}\frac{1}{2\left(x+1\right)}dx results in: \frac{1}{2}\ln\left(2\right). The integral \int_{0}^{1}\frac{-\frac{1}{2}x+\frac{1}{2}}{x^2+1}dx results in: -\frac{1}{4}\ln\left(2\right)+\frac{\pi }{8}.
Final answer to the problem
$\frac{1}{2}\ln\left|2\right|+\frac{\pi }{8}-\frac{1}{4}\ln\left|2\right|$
Exact Numeric Answer
$0.565986$
