1. $\displaystyle \int \frac {1} {t^2 \sqrt {t^2 + 1}} dt $
2. $\displaystyle \int^{\pi}_{\pi/2} \cos(x)^2 \sin(x) dx $
Is this one's result -$\displaystyle \frac {1} {2} $?
Let,Originally Posted by nirva
$\displaystyle u=\cos x$ then, $\displaystyle \frac{du}{dx}=-\sin x$ Thus, you have,
$\displaystyle \int^{\pi}_{\pi/2} \cos^2 x \sin x dx$ becomes,
$\displaystyle -\int^{-1}_{0} u^2 \frac{du}{dx} dx$
Thus,
$\displaystyle \left \int_{-1}^0 u^2 du=\frac{1}{3}u^3 \right| ^0_{-1}$
This one is more complicated.Originally Posted by nirva
This involves trigonomteric substitution.
You have,
$\displaystyle \int \frac{1}{t^2\sqrt{1+t^2}}dt$
Rationalize,
$\displaystyle \int \frac{\sqrt{1+t^2}}{t^2(1+t^2)}dt$
Let,
$\displaystyle u=\tan^{-1}t$
Then,
$\displaystyle \frac{du}{dt}=\frac{1}{1+t^2}$
$\displaystyle \csc u=\frac{\sqrt{1+t^2}}{t}$
$\displaystyle \cot u=\frac{1}{t}$
Thus, express integrand as,
$\displaystyle \int \frac{\sqrt{1+t^2}}{t}\cdot \frac{1}{t} \cdot \frac{1}{1+t^2} dt$
Thus, substitute the previous equations,
$\displaystyle \int \csc u \cot u \frac{du}{dx} dx$
Thus,
$\displaystyle \int \csc u\cot u du$
Now, just determine this new integral.
Use trigonometric identities thus,
$\displaystyle \int \frac{1}{\sin u}\cdot \frac{\cos u}{\sin u} du$
Thus,
$\displaystyle \int \frac{\cos u}{\sin^2u}du$
Let, $\displaystyle x=\sin u$
Then, $\displaystyle \frac{dx}{du}=\cos u$
Express integrand as,
$\displaystyle \int \frac{1}{\sin^2u} \cdot \cos u du$
Make the above substitutions,
$\displaystyle \int \frac{1}{x^2} \frac{dx}{du} du$
Thus,
$\displaystyle \int x^{-2} dx=-\frac{1}{x}+C$
Substitute for $\displaystyle x$ to get,
$\displaystyle -\frac{1}{\sin u}+C$
Thus,
$\displaystyle -\csc u+C$
Substitute for $\displaystyle u$ to get,
$\displaystyle -\csc (\tan^{-1}(t))+C$
Thus,
$\displaystyle -\frac{\sqrt{1+t^2}}{t}+C$
I find the same answer but I don't see why you do the rationalizing and rewriting if the substitution t = tan(u) works immediately.
$\displaystyle
\int {\frac{{dt}}{{t^2 \sqrt {1 + t^2 } }}} \mathop \to \limits_{dt = \sec ^2 u du}^{t = \tan u} \int {\frac{{\sec ^2 udu}}{{\tan ^2 u\sqrt {1 + \tan ^2 u} }}} = \int {\frac{{\cos u}}{{\sin ^2 u}}du}
$
$\displaystyle
\int {\frac{1}{{\sin ^2 u}}d\left( {\sin u} \right) = - \frac{1}{{\sin u}} + C \to } - \frac{1}{{\sin \left( {\arctan t} \right)}} + C
$
Which is the same as your last step, yeilding:
$\displaystyle
-\frac{\sqrt{1+t^2}}{t}+C
$