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How do you use substitution to integrate #6 x e^{4 x^2} dx#?

Answer 1

Ezekiel Alexander

#3/4e^(4x^2)+C#

Let #t=4x^2#, then #dt=8xdx => xdx=dt/8#. So, #int6xe^(4x^2)dx=int6dt/8 e^t=3/4inte^tdt=3/4e^t+C=3/4e^(4x^2)+C#

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Answer 2

Samantha Adkison

To integrate (6xe^{4x^2}) using substitution, let (u = 4x^2). Then, (du = 8x dx). Rearranging, we find (x dx = \frac{1}{8} du). Substituting these into the integral:

[ \int 6xe^{4x^2} dx = \int 6xe^u \frac{1}{8} du ]

Now, the integral becomes:

[ \frac{3}{4} \int e^u du ]

This is a straightforward integral:

[ \frac{3}{4} e^u + C ]

Finally, replacing (u) with (4x^2):

[ \frac{3}{4} e^{4x^2} + C ]

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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.