How Do I Solve These Challenging Definite Integrals?

[EasyStyle4747]

There are several situations I don't know how to solve:

integral of (lnx)/(x^2) dx

integral of e^x(cos2x)dx

integral of ln(1+x^2)dx

I have a quiz tommorow, and after looking through some notes, this is what I couldn't understand.

Please help! thanks.

 

[hypermorphism]

There are several situations I don't know how to solve:

integral of (lnx)/(x^2) dx

integral of e^x(cos2x)dx

integral of ln(1+x^2)dx

I have a quiz tommorow, and after looking through some notes, this is what I couldn't understand.

Please help! thanks.

These can all be done by the appealing to the product rule (integration by parts). Show your work so that we can help with where you're stuck.

 

[Jameson]

$$\int udv = uv - \int{vdu}$$

 

[EasyStyle4747]

Yes, I realize its integration by parts. I've been doing other problems with the product rule. Its just these that I'm stumped on. I basically don't know where to begin with some of these.

for this one: integral of (lnx)/(x^2) dx , are you supposed to change it to (lnx)(x^-2) first and then use parts?

for this one: ntegral of e^x(cos2x)dx, o jeez, i forgot, how do you take the anti-derivative of cos2x

for this one: integral of ln(1+x^2)dx , how do you split it into 2 parts? What do you set u equal to? What do you set dv equal to?

Some general questions:
-Does it matter which part you set dv or u equal to? Or do you HAVE to go in order?

thnx for being helpful people!

 

[t!m]

Order does in fact matter, and generally LIATE (or ILATE) will help you choose what to make 'u' : Logarithmic, Inverse, Algebraic, Trig, Exponential. For example, your first problem:

$$\int \frac{lnx}{x^2}dx$$

let $$u=lnx; dv=\frac{1}{x^2}dx$$
And just derive/integrate to find du and v and plug into the aforementioned expression.

 

[dextercioby]

Those integrals:

a) are not definite.Are indefinite,equivalently,are antiderivarives of certain functions.
b)can be computed via part integration.For the second,two times partial integration is necessary.

 

[Nylex]

for this one: ntegral of e^x(cos2x)dx, o jeez, i forgot, how do you take the anti-derivative of cos2x

The integral of cos 2x = (1/2)sin 2x.

for this one: integral of ln(1+x^2)dx , how do you split it into 2 parts? What do you set u equal to? What do you set dv equal to?

You can write that as 1.ln(1 + x^2) dx, but there's probably a better way of doing that integral.

Some general questions:
-Does it matter which part you set dv or u equal to? Or do you HAVE to go in order?

Yeah, it does because one way round you probably won't be able to do the integral. Usually, you take u to be the part that is "easier" to deal with (eg. polynomials, like x^2) and the dv shouldn't get "harder" to deal with when you integrate (eg. exponentials, trig functions). Say you had the integral of x^2.e^x, you'd take u = x^2 and dv/dx = e^x (and you'd have to do integration by parts twice).