Answers to 4.2.13 all subparts

Since we have already discussed enough problems like this, I'll just give you hints on what to substitute and you'll have to do the rest(which is really simple) by yourself. Ask in the comments if you need any help by the way.

Answer to 4.2.13 (a) :

Substitute, 1 + lnx = t and then dx = x.dt. The integrand will now be as simple as t^(1/3)dt . Integrate it and replace t with 1 + lnx to obtain final answer.

Answer to 4.2.13 (b) :

Substitute, lnx = t then, dx = xdt . Now the integrand become dt/t . Integrate it and you'll get the answer.

Answer to 4.2.13 (c) :

Substitute x^2 = t then dx = dt/2x. Now the integral will look like this :

Now use the standard formula given below to integrate the substituted integrand :

 And then, replace t by x^2 to obtain final answer.

Answer to 4.2.14 (d) :

Use substitution given below : 

Now the integral becomes :

 Use the below standard integration formula to integrate the above integrand :

And after this, replace t with x^n to obtain the final answer.

Answer to 4.2.14 (e) :

Substitute squareroot(x) = t then, dx = 2t.dt
With the above substitutions, the question now has become really simple -- you can integrate it using direct formulas of integration.

Answer to 4.2.14 (f) :

Substitute lnx = t then dx=xdt.
Substitution this the integral will look similar to this :

Now, integrate term by term to obtain the answer.