Final answer to the problem
$-\cos\left(x\right)+\frac{1}{3}\tan\left(3x-1\right)-\frac{1}{2}e^{2x}+4\cdot \left(\frac{1}{\sqrt{2}}\right)\arctan\left(\frac{x}{\sqrt{2}}\right)+C_0$
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Step-by-step Solution
Learn how to solve problems step by step online. Find the integral int(sin(x)+sec(3x-1)^2-e^(2x)4/(x^2+2))dx. Expand the integral \int\left(\sin\left(x\right)+\sec\left(3x-1\right)^2-e^{2x}+\frac{4}{x^2+2}\right)dx into 4 integrals using the sum rule for integrals, to then solve each integral separately. The integral \int\sin\left(x\right)dx results in: -\cos\left(x\right). The integral \int\sec\left(3x-1\right)^2dx results in: \frac{1}{3}\tan\left(3x-1\right). The integral \int-e^{2x}dx results in: -\frac{1}{2}e^{2x}.
Final answer to the problem
$-\cos\left(x\right)+\frac{1}{3}\tan\left(3x-1\right)-\frac{1}{2}e^{2x}+4\cdot \left(\frac{1}{\sqrt{2}}\right)\arctan\left(\frac{x}{\sqrt{2}}\right)+C_0$
