How do you integrate #e^(3x)/(e^6x36)^(1/2)dx#?
Let's assume:
Our integral becomes:
So:
So:
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To integrate the given function (\int \frac{e^{3x}}{\sqrt{e^{6x}36}} dx), use a substitution method:

Substitution: Let (u = e^{6x}  36). Then, differentiate both sides with respect to (x) to find (du).
 (du = 6e^{6x}dx)
This implies we need a factor of (e^{6x}) in the numerator for the substitution to work directly. Notice that we can rewrite the original integral in a form that makes use of this:

Given: (\frac{e^{3x}}{\sqrt{e^{6x}36}} dx)
We rewrite (e^{3x}) as (\frac{e^{6x}}{e^{3x}}) to get: (\frac{\frac{e^{6x}}{e^{3x}}}{\sqrt{e^{6x}36}} dx)

Adjust for (du): We know that (du = 6e^{6x}dx), but we have (\frac{e^{6x}}{e^{3x}}) in our integral, not (6e^{6x}). To compensate, we multiply and divide by 6:

(\int \frac{1}{6} \cdot \frac{6e^{6x}}{e^{3x} \sqrt{e^{6x}36}} dx)
So, we adjust the integral to fit our (du), giving us: (\frac{1}{6} \int \frac{6e^{6x}}{e^{3x} \sqrt{e^{6x}36}} dx)


Substitute and Integrate: Now, substitute (u = e^{6x}  36) and (du = 6e^{6x}dx):
 (\frac{1}{6} \int \frac{du}{\sqrt{u}})

Integrate with Respect to (u):
 The integral of (\frac{1}{\sqrt{u}}) is (2\sqrt{u}), so we get (\frac{1}{6} \cdot 2\sqrt{u} + C) or (\frac{1}{3}\sqrt{u} + C).

Substitute Back: Replace (u) with (e^{6x}  36):
 (\frac{1}{3}\sqrt{e^{6x}  36} + C)
Thus, the integral (\int \frac{e^{3x}}{\sqrt{e^{6x}36}} dx) is (\frac{1}{3}\sqrt{e^{6x}  36} + C).
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When evaluating a onesided 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.
When evaluating a onesided 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.
When evaluating a onesided 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.
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