Question

Evaluate : $\int \frac{x}{\sqrt{x^2+7}}dx$

• A. $\sqrt{x^2+7}+c$
• B. $2\sqrt{x^2+7}+c$
• C. $\frac{2}{\sqrt{x^2+7}}+c$
• D. $sin{h}^{-l}(\frac{x}{\sqrt{7}})+c$

Answer 1

The correct option is A $\sqrt{x^2+7}+c$

Let $x^2+7=t$

Differentiating both sides,

$2xdx=dt$

Substituting the above obtained values back into the expression,

$\frac{xdx}{{(x^2+7)}^{0.5}}$ = $\frac{1}{2}\times \frac{dt}{t^{0.5}}$

$\frac{1}{2}\int \frac{1}{t^{0.5}}dt=\frac{1}{2}\int t^{-0.5}dt$

We know that, $\int x^n.dx=\frac{x^{n+1}}{n+1}$

$=\frac{1}{2}\times 2t^{0.5}+c$

= $t^{0.5}+c$

Substituting $t=x^2+7$ back,

$={(x^2+7)}^{0.5}+c$

$\therefore \int \frac{x}{\sqrt{x^2+7}}dt=(x^2+7{)}^{0.5}+c$

Hence, option 'A' is correct.