Question

Answer this please....
The second pic is the ans for the first question.

Answer 1

Question :

Integrate

$\sf \int \dfrac{4t}{ \sqrt{3t {}^{2} - 7} } dt$

Theory :

•Method of integration

Integration by substitution :

The given integral $\int f(x)dx$ can be transformed into another by changing the independent variable x to t by substituting x = g(t)

Solution :

$\bf \int \dfrac{4t}{ \sqrt{3t {}^{2} - 1 } } dt$

Let $\sf 3t {}^{2} - 1 = p$

$\implies \sf\frac{dp}{dt} = 6t$

$\sf \implies6t dt = dp$

Put this value in Integration, then :

$\sf \int \dfrac{4t}{ \sqrt{3t {}^{2} - 1 } } dt = \int \dfrac{4t}{ \sqrt{p} } \times \dfrac{dp}{6}$

$\sf = \dfrac{4}{6} \int \dfrac{dp}{p {}^{ \frac{1}{2} } }$

$\sf = \dfrac{4}{6} \int p {}^{ \frac{ - 1}{2} } dp$

$\sf = \dfrac{4}{6} ( \dfrac{p {}^{ \frac{ - 1}{2} + 1 } }{ \frac{ - 1}{2} + 1 } )+c$

$\sf = \dfrac{4}{6} \times \dfrac{ \sqrt{p} }{ \frac{1}{2} }+c$

$\sf = \dfrac{8}{6} \sqrt{p}+c$

$\sf = \dfrac{4}{3} \sqrt{3t {}^{2} - 7 } +c$

it is the required solution!