# Examples of Integration by Substitution

Example: Evaluate the integral

with respect to $x$.

We have integral

Putting $t = \ln \ln x$ and differentiating $dt = \frac{1}{{x\ln x}}dx$

Now the above integral of the form

We observe that the derivation of given function is in the given problem, so using the general power formula of integration

Here $f\left( t \right) = \sin x$ implies that $f'\left( t \right) = \cos x$

Now using the original substitution again $t = \ln \ln x$ 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 differentiating $dt = \frac{1}{{2\sqrt {x + 1} }}dx$ implies $2dt = \frac{1}{{\sqrt {x + 1} }}dx$

Now by using these values, equation (i) becomes

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