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^{3/2} , 4\leq x\leq 6\\ \end{matrix}\right.

 

17 Mar 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

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