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How do you find the integral of #(1 + e^2x) ^(1/2)#?

Answer 1

#int((1+e^2x)^(1/2))dx=2/(3e^2) (1+e^2x)^(3/2) #

Integrate by method of substitution.

Solution: (1) Let u = #sqrt(1+e^2x# (2) Take the square of u, hence, #u^2=1+e^2x# (3) Take the derivative of both sides, hence, #2udu=e^2dx# (4) Substitute 'u' and 'du' to the original differential eqn. #int u * 2u/e^2 du# (5) Integrate, #2/e^2intu^2du=2/(3e^2)u^3# (6) Replace 'u' in terms of 'x' by using the defined value of 'u' #int((1+e^2x)^(1/2))dx=2/(3e^2) (1+e^2x)^(3/2) #

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Answer 2

Charles Anctil

To find the integral of (1 + e^(2x))^(1/2), you can use trigonometric substitution. Let u = e^x, then du = e^x dx. Substituting u and du into the integral, we get:

∫(1 + e^(2x))^(1/2) dx = ∫(1 + u^2)^(1/2) * (1/u) du.

Now, you can use trigonometric substitution or other methods to evaluate this integral. One common approach is to use a trigonometric substitution, where you let u = tan(theta), then du = sec^2(theta) d(theta). After substitution, you will have an integral in terms of trigonometric functions, which you can then solve.

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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.