Integration Question

Hi. I am slightly confused regarding integration. I know you can integrate, for example (2x+4)^3 but adding one to the power, dividing by this new power, and then dividing by the derivative of the bracket, but why can't you do this with, for example (sinx)^2 or (lnx)^2 ?. I couldn't integrate the latter of these correctly on a recent paper and I am still confused. Thanks.

Reply 1

mqb2766

21

Original post by Har6547

Hi. I am slightly confused regarding integration. I know you can integrate, for example (2x+4)^3 but adding one to the power, dividing by this new power, and then dividing by the derivative of the bracket, but why can't you do this with, for example (sinx)^2 or (lnx)^2 ?. I couldn't integrate the latter of these correctly on a recent paper and I am still confused. Thanks.

For the integral of
(2x+4)^3
youre doing substitution or the reverse chain rule, so substitution is
u = 2x+4
du = 2 dx
and the integral becomes
1/2 Int u^3 du
....
A suitable (obvious?) substitution has made the integration problem easier.

sin(x)^2 would probably be easier by parts or a trig identity to a double angle. Does substitution make it easier? Similarly for ln(x)^2. When integrand involve ln(x), the "sneaky" by parts approach is usually to choose the function you integrate to be 1 and the function you differentiate to be the original integrand, so by parts on
Int 1 * ln^2(x)

With integration, you have to be prepared to try a line or two of substition, by parts, identity transformation ... and see which one makes the problem easier.

(edited 1 year ago)

Reply 2

davros

16

Original post by Har6547

Hi. I am slightly confused regarding integration. I know you can integrate, for example (2x+4)^3 but adding one to the power, dividing by this new power, and then dividing by the derivative of the bracket, but why can't you do this with, for example (sinx)^2 or (lnx)^2 ?. I couldn't integrate the latter of these correctly on a recent paper and I am still confused. Thanks.

Your first example is a very, very special case where you have a function ("raising to the 3rd power") of a linear function (2x+4). Unless you're applying a function to a linear expression you can't do what you're attempting - it just isn't valid!

You need to use a valid method of integration such as substitution, or integration by parts. In the case of (sin x)^2 there is a much easier way which involves rewriting (sin x)^2 in terms of a double angle formula :smile:

Reply 3

Har6547

OP

13

Original post by mqb2766

For the integral of
(2x+4)^3
youre doing substitution or the reverse chain rule, so substitution is
u = 2x+4
du = 2 dx
and the integral becomes
1/2 Int u^3 du
....
A suitable (obvious?) substitution has made the integration problem easier.

sin(x)^2 would probably be easier by parts or a trig identity to a double angle. Does substitution make it easier? Similarly for ln(x)^2. When integrand involve ln(x), the "sneaky" by parts approach is usually to choose the function you integrate to be 1 and the function you differentiate to be the original integrand.

With integration, you have to be prepared to try a line or two of substition, by parts, identity transformation ... and see which one makes the problem easier.

Ok thank you for your reply. So could substitution be used for (lnx)^2? Like u=lnx ? Or would you just not do this because it is too complex?

Reply 4

mqb2766

21

Original post by Har6547

Ok thank you for your reply. So could substitution be used for (lnx)^2? Like u=lnx ? Or would you just not do this because it is too complex?

You really need to get in the habit of trying it and seeing if it works.
Post what you get for the transformed integral.

Reply 5

davros

16

Original post by Har6547

Ok thank you for your reply. So could substitution be used for (lnx)^2? Like u=lnx ? Or would you just not do this because it is too complex?

As mqb says, try it and see - it really is the best way to get confident with integration! So if u = ln x, what is x in terms of u? What does dx become in terms of du? What does your x-integral transform into as a u-integral? Is the resulting integral something you can apply one of the standard rules (further substitution or IBP) to ?

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