IGNOU MST 1 2018 - Evaluate the following integrals: i) ii)

Question

Evaluate the following integrals:

i) $\int \frac{3x}{(x^{2}+k^{2})^5} dx$

ii) $\int_{0}^{6}f(x)dx where$ $f(x) = \left\{\begin{matrix} x^2 + 3, &0\leq x< 3 \\ 2\sqrt{x},&3\leq x< 4 \\ e^{-2x} - x\tfrac{3}{2},& 4\leq x\leq 6 \end{matrix}\right.$

16 Feb 2018

Answer :

Word Count : 189

$i)\int \frac{3x}{(x^2 + k^2)^5}dx$

Let

$x^{2}+k^{2}= m$

$\Rightarrow x=\sqrt{m-k^{2}}$

$\Rightarrow 2x \frac{dx}{dm} = 1$

$\Rightarrow \frac{dx}{dm} = \frac{1}{2x}$

$\Rightarrow dx = \frac{1}{2x} dm$

$= \int \frac{3x}{m^{5}} \times \frac{1}{2x} dm$

$\therefore eqution (1) becomes$

$= \int \frac{3}{2m^5} dm$ _________ ____ ______ ______ ___ _____ _________ ____ ________.
____ ________ ___ ___ __________.
_________ _______ ______ _______ ___ ______ ____ ______.
_________ _____ ____ ____ _________ ______ _________ _________ ________.
_____ ____ _________ _______ ________ ______ ________ __________ _________ ________.
________ __________ ____ _______ ___ _____.
___ ________ _____ _________ ____ _____ __________ ________ ___ __________ ___ __________.
_________ _____ __________ ______ _________ ___ ________ _____ __________ __________ ____ _______.
_______ _________ ____ _____ _______ ________.
__________ _______ _______ _________ __________.
______ _________ ______ _________ ______ ______ ___ ______.
_____ ____ ____ _______ ___ ________ _______ _________ ______.
_________ _________ ____ _____ __________ ________ __________ _______ __________.
___ _________ __________ ___ ____ _______ ___ __________ _______.
___ __________ ________ _______ ________ _________.
_______ __________ ______ ______.