Final answer to the problem
$\frac{1}{2}x^3e^{2x}+\frac{5}{2}x^2e^{2x}-\frac{11}{8}e^{2x}+\frac{3}{4}xe^{2x}+\left(-\frac{3}{4}\right)x^2e^{2x}-5x^2+C_0$
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Step-by-step Solution
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1
Simplify the expression
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$\frac{1}{2}\left(x^3+5x^2-2\right)e^{2x}-\frac{3}{2}\int e^{2x}x^2dx-10\int xdx$
Learn how to solve problems step by step online. Find the integral (x^3+5x^2+-2)1/2e^(2x)-int(1/2e^(2x)*3x^2+10x)dx. Simplify the expression. Multiply the single term -\frac{3}{2} by each term of the polynomial \left(\frac{1}{2}x^2e^{2x}-\frac{1}{2}xe^{2x}+\frac{1}{4}e^{2x}\right). The integral -\frac{3}{2}\int e^{2x}x^2dx results in: \frac{1}{2}\cdot \left(-\frac{3}{2}\right)x^2e^{2x}-\frac{1}{2}\cdot \left(-\frac{3}{2}\right)xe^{2x}+\frac{1}{4}\cdot \left(-\frac{3}{2}\right)e^{2x}. Gather the results of all integrals.
Final answer to the problem
$\frac{1}{2}x^3e^{2x}+\frac{5}{2}x^2e^{2x}-\frac{11}{8}e^{2x}+\frac{3}{4}xe^{2x}+\left(-\frac{3}{4}\right)x^2e^{2x}-5x^2+C_0$
