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How can I evaluate the integral #intx^2e^(4x)dx#?

Answer 1

Charles Anctil

Where you have: I would use integration by parts.

#intf(x)*g(x)dx=F(x)*g(x)-intF(x)*g'(x)dx#

Where:

#F(x)=intf(x)dx#

#g'(x)# is the derivative of #g(x)#

I'll decide on:

#f(x)=e^(4x)# and #g(x)=x^2#

When you integrate, you obtain:

#intx^2e^(4x)dx=x^2e^(4x)/4-int2xe^(4x)/4dx=# #=x^2e^(4x)/4-intxe^(4x)/2dx=# #=x^2e^(4x)/4-1/2intxe^(4x)dx=# and by parts again: #=x^2e^(4x)/4-1/2[xe^(4x)/4-int1e^(4x)/4dx]=# #=x^2e^(4x)/4-xe^(4x)/8+e^(4x)/32+c=# #=e^(4x)(x^2/4-x/8+1/32)+c#

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Answer 2

Adam Adkins

To evaluate the integral ∫x^2e^(4x)dx, you can use integration by parts. Let u = x^2 and dv = e^(4x)dx. Then, differentiate u to find du and integrate dv to find v. After that, apply the integration by parts formula:

∫u dv = uv - ∫v du

Once you've calculated the values of u, dv, du, and v, substitute them into the formula and evaluate the integral.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.