A question about Indefinite trigonometric integration

Question

I am practising a unit on Integration. I am going through some past year papers, and there are some types of questions that I could not solve. So if anyone could help me in this, I'd really appreciate this.

Evaluate the following integrals: $$ i)\quad\int\left(\frac{2}{\sqrt{x}}+2e^{-4x}+\frac{1}{3(1-x)}\right)\:\mathrm{d}x $$

In this, I managed to get to a point where the answer is: $$ 4\sqrt{x}-\frac{e^{-4x}}{2}+\frac{\ln|x+1|}{3}+C $$

Is this the final answer or there's more I can do here. I'm specially confused about the $|x+1|$ part.

Next is a trigonometric substituition question. I've tried basic ones of this type, but this one is very difficult and complicated for me. If someone could point me into a direction then maybe I can try solving.

Show that: $$ \int\frac{x^2}{\sqrt{36-x^2}}\:\mathrm{d}x=18\sin^{-1}\left(\frac{x}{6}\right)-\frac{1}{2}x\sqrt{36-x^2}+C $$ with an appropriate trigonometric substitution.

Answer 1

• i)

The result has to be slightly modified; the last function is $-\frac{1}{3}\ln|1-x|$; in fact its derivative w.r.t. $x$ is equal to

$$-\frac{1}{3}\frac{-1}{1-x}=\frac{1}{3(1-x)}, $$

if $1-x>0$ and

$$-\frac{1}{3}\frac{1}{x-1}=\frac{1}{3(1-x)}, $$

if $1-x<0$.

Moreover, the result of the integral is given by the sum of the proposed functions (with the replacement of the third with the one above) plus an arbitrary constant $C$.

• ii)

With the substitution $x=6\sin t$, $dx=6\cos t dt$, the proposed integral is equal to

$$\int \frac{36\sin^2 t}{6\sqrt{1-\sin^2 t}}6\cos t dt=\int \frac{36\sin^2 t}{|\cos t|}\cos t dt= \int36\sin^2 t dt, $$

as $\cos t> 0$, if $t\in(-\frac{\pi}{2},\frac{\pi}{2})$, which is the taken image of $\operatorname{arcsin}(\cdot)$.

All you need is

$$36\int\sin^2 t dt=36\left(\frac{1}{2}t-\frac{1}{4}\sin(2t)\right)+C,$$

with $\sin(2t)=2\sin t\cos t$.

Answer 2

Here we can put: $$ x=6\sin\theta,\:\text{then we can get}\;\frac{\mathrm{d}x}{\mathrm{d}\theta}=6\cos\theta\:\mathrm{d}\theta $$ so that your integral becomes: $$ \int\frac{36\sin^2\theta(6\cos\theta)}{6\cos\theta}\:\mathrm{d}\theta=\int 36\sin^2\theta\:\mathrm{d}\theta $$ I think, from this you should be able to proceed further.

EDIT: For your first question, yes that is the final answer except the fact that you made a small mistake. Always to check your answer, differentiate it and see if it's the same as your question.

ANS: $$ 4\sqrt{x}-\frac{e^{-4x}}{2}+\frac{\ln|1-x|}{3}+C $$