Evaluate the integrals [closed]

Question

I have a few integrals here that I need help on, plz give me some hints or suggestions on how to solve them. Thanks so much!

1. $$\int \frac{7\ln x}{x\sqrt{2 + (\ln x)^2}}\ dx$$
2. $$\int \sqrt{\frac{9+x}{9-x}}\ dx$$
3. $$\int 4\sqrt{4 + e^x}\ dx$$
4. $$\int \frac{dx}{x\sqrt{8x + 49}}$$

Answer 1

Hint on the first one: try $u=\sqrt{2+(\log x)^2}$. Then $u^2=2+(\log x)^2$, $2u\,du=2(\log x)/x$, etc.

Answer 2

HINT : The first one is straightforward if you use $\ln x = u$.

That is $$\int \frac{7\ln x}{x\sqrt{2 + (\ln x)^2}}\ dx=\int \frac{7u}{\sqrt{2 + u^2}}\ du=7\int (\sqrt{2 + u^2})'\ du=7 \sqrt{2 + u^2}=7 \sqrt{2 +(\ln x)^2}+C$$

Answer 3

1. $\frac{7}{2}\int \frac{d[2+(\ln x)^2]}{\sqrt{2+(\ln x)^2}}=\frac{7}{2}\int\frac{du}{\sqrt{u}}=...$

2. Substitute $x=9\cos2\theta$

3. Substitute $4+e^x=y^2$

4. $=\frac{8}{3}\int ue^{-u} du=... $

5. Substitute $\sqrt{8x+49}=y$

Answer 4

First of all I apologize that my mathJax is not rendering, and my formula may fail.

I'd say by attempting $u$-substitution, the problem could be solved.

== Why $u$-substitute? ==

Then, why $u$-substitute? To change a strange-looking integrand to something you might recognize. Even if obscured at the beginning, try to substitute different expressions (either schematically or randomly). You may stumble upon the right answer.

Take this little exercise:

$$\int \frac{\cos x}{1+\sin x \,\mathrm d x}$$

There is a fraction. The denominator is the sum of $1$ and a function $\sin x$.

If you are experienced enough to notice that the derivative of $1+\sin x$ is $\cos x$, then you may simply let

$$u=1+\sin x$$ with $$\,\mathrm d u = \cos x \,\mathrm d x$$ so that

$$\int \frac{\cos x}{1+\sin x \,\mathrm d x} = \int\frac{du}{u}=\ln\left|u\right|+C$$

Well, you say where's the x? Using careful manipulation we chose to hide x by letting $u=1+\sin x$ there in the beginning. Now, it's high time that we substituted the $u$ into the R.H.S. of the last equation and obtain

$$\ln\left|1+\sin x\right| + C.$$

(example from book: A First Course in Mathematical Modelling, F. R. Giodano)

== Try different substitutions ==

The third integral of OP is not to easy at first. OP would try $u=e^x$, ($\,\mathrm d x = u^{-1} \,\mathrm d u$) but it doesn't make things easier. Try to put more things into your substitution, try a few more minutes please.

Spoiler alert!

Try to simplify the square root.

Spoiler alert!

Try $u=4+e^x$

Spoiler alert! Finish the first few spoilers and proceed.

Try integration by parts.

Alternatively, try to differentiate $u^{3/2} \ln u$, and see what terms you need to add.