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Calculus/Integration by parts

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Expert: Alon Mandes - 12/9/2009


Question
Hello. I need to find the integral of 3e^(5x)* sin(2x) dx. I have been having a lot of trouble with choosing a u and a dV and I get all of them confused. I also don't understand why this integral seems to go on forever. I must be missing something....

Thanks,
Kristin

Answer
In this question we will use the Integration by parts twice in cycle way . Let's do it :
We take u=e^(5x) & v'=sin(2x). This will give us :
∫3e^(5x)* sin(2x)dx=(-3/2)e^(5x)cos(2x) - ∫(-15/2)e^(5x)cos(2x)dx . This time we take :
u=15e^(5x) & v'=(-1/2)cos(2x) . Then we get :
∫3e^(5x)* sin(2x)dx=(-3/2)e^(5x)cos(2x) - [15e^(5x)sin(2x) - ∫75e^(5x)sin(2x)dx] .
Here's the trick : If we set the term ∫e^(5x)* sin(2x)dx as I , then we get :
3I=(-3/2)e^(5x)cos(2x) - [15e^(5x)sin(2x) - 75I]
3I= (-3/2)e^(5x)cos(2x) - 15e^(5x)sin(2x) - 75I
72I=(-3/2)e^(5x)cos(2x) - 15e^(5x)sin(2x) . Thus,
I=(1/72)(-3/2)*e^(5x)cos(2x) - (1/72)*15*e^(5x)sin(2x) .

Alon.

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Alon Mandes

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Kind of questions I can answer : Limits, Derivatives, Integration, Implicit functions, continuousity, differentiation ,Extremum problems, Lagrange multipliers, Gradients, Surface integrals, Multi variables functions ,Multi variables Integrals,Complex variables ,Complex functions, Curves, Trajectory integrals & Vector analyse,Divergence,Rotor & word problems. Kind of question I can't answer : Economics,Combinatorics,infinite series & convergence ,Statistics & Probabilities .

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1. I'm a team member of mathnerds (math site for answering questions) 2. I'm a team member in the Student's Union of the Technion, helping students who have problems in mathematics. 3. 2 years of experience as a math teacher in college. 4. I give free homework help for high school students in Mathematics & Physics. 5. I teach part time in collage the subjects : "Digital Signal Processing" , "Random Signals & Noise" , "Complex Functions".

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M.A in Mathematics & Bs.c in Electronics.

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