9–40. Integration by parts Evaluate the following integrals us...

Question

9–40. Integration by parts Evaluate the following integrals using integration by parts.

36. ∫ from 0 to ln2 x eˣ dx

Accepted Answer

Welcome back, everyone. Evaluate the following integral using integration by parts. Integral from 0 to a line of 3 of X multiplied by E raises the power of XD X. For this problem, let's first focus on the indefinite integral X multiplied by E erase the power of XD X. We're going to use integration by parts. Let's recall that the integral of UDV is equal to UV minus the integral of VDU. What we're going to do in this problem is simply set U equals X. So we can now show that the EU is DEX, right? And then DV is going to be E erase the power of XDX, right? So E erase the power of XDX. If we integrate both sides, we're going to get The integral of DV, which is V. is equal to the integral of EXDX which is E erase the power of X, right? We don't need to include a constant of integration C. So we now have V and U, right? According to the integration by parts, we get integral of. X multiplied by ease the power of XD X is equal to UV, so that would be X multiplied by E erase the power of X minus the integral of VDU. So the integral of Ea the power of X, and DU is D X. We can continue this equation and show that this is equal to x. Ease the power of x minus the integral of E erase the power of X E erase the power of X, and we're going to add a constant of integration C. So this is for an indefinite integral. We can also factor it out and show that this is equal to E raise the power of X and C X minus 1 plus a constant of integration C. So now if we have a definite integral from 0 to LN of 3. Of X multiplied by ease the power of XD X. This is going to be E erase the power of X multiplied by X minus 1. We're using our previous result evaluated from 0 to LN of 3. Now we're not using a constant of integration C because this is a definite integral. And now using the evaluation theorem, we're going to get E raise to the power of LN of 3. Multiplied by LN of 3 minus 1. Minus E raises the power of 0 multiplied by 0 minus 1, right? And now we simply want to simplify. Ease the power of LN of 3 is 3, so we get 3. In C's of 3 minus 1 minus E raises the power of 0 is a 1 and 1 multiplied by -1 is -1. So minus -1 is basically + 1. Now distributing, we get 3 LN of 3 minus 3 + 1, so that would be. -2. Our final answer is 3 of 3 minus 2. Thank you for watching.

9–40. Integration by parts Evaluate the following integrals using integration by parts.
36. ∫ from 0 to ln2 x eˣ dx

Key Concepts

Integration by Parts

Definite Integrals

Exponential and Logarithmic Functions

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