Final answer to the problem
$\frac{1}{3}x^3e^{3x}-\frac{1}{3}x^{2}e^{3x}+\frac{2}{9}xe^{3x}-\frac{2}{27}e^{3x}+C_0$
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Step-by-step Solution
Learn how to solve integrals of exponential functions problems step by step online. Find the integral int(x^3e^(3x))dx. We can solve the integral \int x^3e^{3x}dx by applying the method of tabular integration by parts, which allows us to perform successive integrations by parts on integrals of the form \int P(x)T(x) dx. P(x) is typically a polynomial function and T(x) is a transcendent function such as \sin(x), \cos(x) and e^x. The first step is to choose functions P(x) and T(x). Derive P(x) until it becomes 0. Integrate T(x) as many times as we have had to derive P(x), so we must integrate e^{3x} a total of 4 times. With the derivatives and integrals of both functions we build the following table.
Final answer to the problem
$\frac{1}{3}x^3e^{3x}-\frac{1}{3}x^{2}e^{3x}+\frac{2}{9}xe^{3x}-\frac{2}{27}e^{3x}+C_0$
