Use the reduction formulas in a table of integrals to evaluate the following integrals. int x^3e^{2x}dx

Use the reduction formulas in a table of integrals to evaluate the following integrals. int x^3e^{2x}dx

Question
Integrals
asked 2020-11-09
Use the reduction formulas in a table of integrals to evaluate the following integrals.
\(\int x^3e^{2x}dx\)

Answers (1)

2020-11-10
\(\text{Given }I=\int x^3e^{2x}dx\)
\(\text{For evaluating given integral, we use integral by parts theorem}\)
\(\text{According to integral by parts theorem}\)
\(\int f(x)g'(x)dx=f(x)\int g'(x)dx-[\int f'(x)\int g'(x)dx)dx]\)
\(\text{Here, }f(x)=x^3,g'(x)=e^{2x}\)
\(\text{So, be using equation}\)
\(I=\int x^3e^{2x}dx\)
\(=x^3\int e^{2x}dx-[\int(\frac{d}{dx}(x^3)\int e^{2x}dx)dx]\)
\((\because \int e^{kx}dx=\frac{e^{kx}}{k}+c,\frac{d}{dx}(x^n)=(nx^{n-1})\)
\(=x^3(\frac{e^{2x}}{2})-[\int (3x^2)(\frac{e^{2x}}{2}dx]\)
\(=\frac{x^3e^{2x}}{2}-\frac{3}{2}[\int x^2e^{2x}dx]\)
\(=\frac{x^3e^{2x}}{2}-\frac{3}{2}[x^2\int e^{2x}dx-[\int[\frac{d}{dx}(x^2)\int e^{2x}dx]dx]]\)
\(=\frac{x^3e^{2x}}{2}-\frac{3}{2}[x^2(\frac{e^{2x}}{2})-[\int(2x)\frac{e^{2x}}{2}dx]]\)
\(=\frac{x^3e^{2x}}{2}-\frac{3x^2e^{2x}}{4}+\frac{3}{2}\int xe^{2x}dx\)
\(=\frac{x^3e^{2x}}{2}-\frac{3x^2e^{2x}}{4}+\frac{3}{2}[x\int e^{2x}dx-[\int(\frac{d}{dx}(x)\int e^{2x}dx)dx]]\)
\(=\frac{x^3e^{2x}}{2}-\frac{3x^2e^{2x}}{4}+\frac{3}{2}[x(\frac{e^{2x}}{2}-[\int(1)(\frac{e^{2x}}{2})dx]]\)
\(=\frac{x^3e^{2x}}{2}-\frac{3x^2e^{2x}}{4}+\frac{3xe^{2x}}{4}-\frac{3e^{2x}}{8}+c\)
\(=\frac{(4x^3-6x^2+6x-3)e^{2x}}{8}+c\)
\(\text{Hence, given integral is equal to } \frac{(4x^3-6x^2+6x-3)e^{2x}}{8}+c\)
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