Let 0. If ^x x x + - x dx = 16 + 20 215, then is equal to: — Solution | JEE Mathematics

MCQMediumJEE 2023Indefinite Integrals

Let $\alpha > 0$ . If $\int_{\alpha}^{x} \frac{x}{\sqrt{x + \alpha - \sqrt{x}}} \, dx = \frac{16 + 20 \sqrt{2}}{15},$ then $\alpha$ is equal to:

Answer

Correct answer:A

Standard Method

Given:

\int_{0}^{\alpha}\frac{x}{\alpha}\left(\sqrt{x+\alpha}+\sqrt{x}\right)dx = \frac{16 + 20\sqrt{2}}{15}

Find: $\alpha$

After rationalising,

\int_{0}^{\alpha}\frac{x}{\alpha}\left(\sqrt{x+\alpha}+\sqrt{x}\right)dx

= \int_{0}^{\alpha}\frac{1}{\alpha}\left[(x+\alpha)^{3/2}-\alpha(x+\alpha)^{1/2}+x^{3/2}\right]dx

Integrating term by term,

= \frac{1}{\alpha}\left[\frac{2}{5}(x+\alpha)^{5/2}-\alpha\frac{2}{3}(x+\alpha)^{3/2}+\frac{2}{5}x^{5/2}\right]_{0}^{\alpha}

Substituting the limits,

= \frac{1}{\alpha}\left[\frac{2}{5}(2\alpha)^{5/2}-\frac{2\alpha}{3}(2\alpha)^{3/2}+\frac{2}{5}\alpha^{5/2}-\frac{2}{5}\alpha^{5/2}+\frac{2}{3}\alpha^{5/2}\right]

= \frac{1}{\alpha}\left[\frac{2^{7/2}\alpha^{5/2}}{5}-\frac{2^{5/2}\alpha^{5/2}}{3}+\frac{2}{3}\alpha^{5/2}\right]

So,

= \alpha^{3/2}\left[\frac{2^{7/2}}{5}-\frac{2^{5/2}}{3}+\frac{2}{3}\right]

= \frac{\alpha^{3/2}}{15}(24\sqrt{2}-20\sqrt{2}+10)

= \frac{\alpha^{3/2}}{15}(4\sqrt{2}+10)

Now equating with the given value,

\frac{\alpha^{3/2}}{15}(4\sqrt{2}+10)=\frac{16+20\sqrt{2}}{15}

Hence,

\alpha = 2

Therefore, the correct option is A.

Answer from extracted working

The solution explicitly concludes:

\alpha = 2

and states that the correct option is A. Therefore, the answer is A.

Using the incorrect integrand directly from the printed expression without following the rationalisation shown in the solution. This changes the algebraic form and leads to a wrong value of $\alpha$ . Follow the transformed integrand exactly as obtained after rationalising.
Making errors while evaluating the limits at $x=0$ and $x=\alpha$ . Missing the cancellation of terms in $\alpha^{5/2}$ produces an incorrect constant factor. Substitute both limits carefully before simplifying.
Simplifying powers such as $(2\alpha)^{5/2}$ or $(2\alpha)^{3/2}$ incorrectly. The exponents apply to both $2$ and $\alpha$ , so write them as $2^{5/2}\alpha^{5/2}$ and $2^{3/2}\alpha^{3/2}$ before combining terms.

Get unlimited AI-adaptive practice, mastery tracking, and an AI tutor that explains every step — free to start.