x" sin ax cos ax dx sin ax dr, for a + 0 a

Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter10: Sequences, Series, And Probability
Section10.8: Probability
Problem 18E
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Reduction formulas Use integration by parts to derive the following reduction formulas.

x" sin ax
cos ax dx
sin ax dr, for a + 0
a
Transcribed Image Text:x" sin ax cos ax dx sin ax dr, for a + 0 a
Expert Solution
Step 1

We have to use integration by parts to derive the reduction formula:

xncosaxdx=xnsinaxa-naxn-1sinaxdx, for a0.

When two function is in product then we do use integration by parts method

in this method we decide u(first function) and v (second function)

according to ILATE order rule,

Here ILATE stands for

Inverse function

Logarithmic function

Algebraic function

Trigonometric function

Exponential function 

formula:

uvdx=uvdx-dudxvdxdx

So taking,

u=xnv=cosax

We know the integration:

cosax=sinaxa+c

 

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