Question

integral of dx/x+3√x


​

Answer 1

GIVEN :–

$\\ \implies \rm \: \: A\:\: Function \: \: \dfrac{dx}{x + 3 \sqrt{x} }$

TO FIND :–

• Integration of function = ?

SOLUTION :–

• Let the function –

$\\ \implies \rm I = \int \dfrac{dx}{x + 3 \sqrt{x} }$

• We should write this as –

$\\ \implies \rm I = \int \dfrac{dx}{\sqrt{x}( \sqrt{x} + 3)}$

• Now put $\rm \sqrt{x} = t$ –

• Differentiate with respect to 'x' –

$\\ \rm \implies \dfrac{d(\sqrt{x})}{dx} = t$

$\\ \rm \implies \dfrac{1}{2 \sqrt{x}}dx = dt$

$\\ \rm \implies \dfrac{1}{ \sqrt{x}}dx =2dt$

• So that –

$\\ \implies \rm I = \int \dfrac{2dt}{(t+3)}$

$\\ \implies \rm I = 2 \int \dfrac{dt}{(t+3)}$

$\\ \implies \rm I = 2 \ln(t+3) + c$

• Now replace 't' –

$\\ \implies \large{ \boxed{ \rm I = 2 \ln( \sqrt{x} +3) + c}}$