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

shadsiei 2020-11-09 Answered
Use the reduction formulas in a table of integrals to evaluate the following integrals.
x3e2xdx
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Expert Answer

izboknil3
Answered 2020-11-10 Author has 99 answers
Given I=x3e2xdx
For evaluating given integral, we use integral by parts theorem
According to integral by parts theorem
f(x)g(x)dx=f(x)g(x)dx[f(x)g(x)dx)dx]
Here, f(x)=x3,g(x)=e2x
So, be using equation
I=x3e2xdx
=x3e2xdx[(ddx(x3)e2xdx)dx]
(ekxdx=ekxk+c,ddx(xn)=(nxn1)
=x3(e2x2)[(3x2)(e2x2dx]
=x3e2x232[x2e2xdx]
=x3e2x232[x2e2xdx[[ddx(x2)e2xdx]dx]]
=x3e2x232[x2(e2x2)[(2x)e2x2dx]]
=x3e2x23x2e2x4+32xe2xdx
=x3e2x23x2e2x4+32[xe2xdx[(ddx(x)e2xdx)dx]]
=x3e2x23x2e2x4+32[x(e2x2[(1)(e2x2)dx]]
=x3e2x23x2e2x4+3xe2x43e2x8+c
=(4x36x2+6x3)e2x8+c

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