Final answer to the problem
$\frac{1}{2}\ln\left|4x^2+1\right|+7\cdot \left(\frac{1}{\sqrt{5}}\right)\arctan\left(\frac{x}{\sqrt{5}}\right)-3\ln\left|\sqrt{x^2+5}\right|+C_1$
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Step-by-step Solution
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1
Simplify the expression
$4\int\frac{x}{4x^2+1}dx+\int\frac{-3x+7}{x^2+5}dx$
Learn how to solve integration techniques problems step by step online.
$4\int\frac{x}{4x^2+1}dx+\int\frac{-3x+7}{x^2+5}dx$
Learn how to solve integration techniques problems step by step online. Integrate int((4x)/(4x^2+1)+(-3x+7)/(x^2+5))dx. Simplify the expression. The integral 4\int\frac{x}{4x^2+1}dx results in: \frac{1}{2}\ln\left(4x^2+1\right). The integral \int\frac{-3x+7}{x^2+5}dx results in: 3\ln\left(\frac{\sqrt{5}}{\sqrt{x^2+5}}\right)+7\cdot \left(\frac{1}{\sqrt{5}}\right)\arctan\left(\frac{x}{\sqrt{5}}\right). Gather the results of all integrals.
Final answer to the problem
$\frac{1}{2}\ln\left|4x^2+1\right|+7\cdot \left(\frac{1}{\sqrt{5}}\right)\arctan\left(\frac{x}{\sqrt{5}}\right)-3\ln\left|\sqrt{x^2+5}\right|+C_1$
