# Examples of Integration by Substitution

Example: Evaluate the integral

with respect to $x$

We have integral

Putting $t = \ln \ln x$ and differentiate $dt = \frac{1}{{x\ln x}}dx$
Now the above integral of the form, we have

We observe that derivation of given function is in the given problem, so using general power formula of integration, we have
Here $f\left( t \right) = \sin x$ implies that $f'\left( t \right) = \cos x$
Using the formula

Now putting again the original substitution $t = \ln \ln x$ in the result of the integration, we have

Example: Integrate $\frac{{{e^{\sqrt {x + 1} }}}}{{\sqrt {x + 1} }}$ with respect to $x$.

Consider the function to be integrate

Putting $t = \sqrt {x + 1}$ and differentiate $dt = \frac{1}{{2\sqrt {x + 1} }}dx$ implies $2dt = \frac{1}{{\sqrt {x + 1} }}dx$
Now equation (i) becomes, by putting the values

Using the formula of integration $\int {{e^x}dx = {e^x} + c}$, we have