   Chapter 8.2, Problem 75E

Chapter
Section
Textbook Problem

Proof In Exercises 71-76, use integration by parts to prove the formula. (For Exercises 71-74, assume that n is a positive integer.) ∫ e a x sin   b x   d x = e a x ( a sin b x + b cos b x ) a 2 + b 2 + C

To determine

To prove: The value of the given integral is eaxsinbxdx=eax(asinbx+bcosbx)a2+b2+C

Explanation

Formula used:

By using Integration by parts

uv'=uvu'v(1)

Proof:

Consider the given formula.

eaxsinbxdx=eax(asinbx+bcosbx)a2+b2+C

Take the left hand side of the equation mentioned above

We have.

eaxsinbxdx

Now, Apply the integration by part rule mentioned above.

uv'=uvu'v(1)

Let, u=sinbx

On Differentiating with respect to x.

We get,

u'=bcos(bx)

Let, dv=eaxdx

Now, Integrate the equation.

We get,

v=eaxa

On substituting u=sinbx, u'=bcos(bx) and u'=bcos(bx) in equation (1)

we get

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