The value of the integral ∫0121+3x+121-x614 dx is

JEE Advanced · Mathematics · 27. Definite Integration

The value of the integral $\int_{0}^{\frac{1}{2}} \frac{1 + \sqrt{3}}{{({(x + 1)}^{2} {(1 - x)}^{6})}^{\frac{1}{4}}} d x$ is

A 4
B 26
C 9
D 2

Verified Solution

Answer & Solution

Correct Answer

(D) 2

Step-by-step Solution

Detailed explanation

\int_{0}^{\frac{1}{2}} \frac{(1 + \sqrt{3}) d x}{{[{(1 + x)}^{2} {(1 - x)}^{6}]}^{\frac{1}{4}}}

\int_{0}^{\frac{1}{2}} \frac{(1 + \sqrt{3}) d x}{{(1 + x)}^{2} {[\frac{{(1 - x)}^{6}}{{(1 + x)}^{6}}]}^{\frac{1}{4}}}

Put

\frac{1 - x}{1 + x} = t \Rightarrow \frac{- 2 d x}{{(1 + x)}^{2}} = d t

I = \int_{1}^{\frac{1}{3}} \frac{(1 + \sqrt{3}) d t}{- 2 t^{\frac{6}{4}}} = \frac{- (1 + \sqrt{3})}{2} \times {|- \frac{2}{\sqrt{t}}|}_{1}^{\frac{1}{3}} = (1 + \sqrt{3}) (\sqrt{3} - 1) = 2

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

Same subject

Let $f: R \rightarrow R$ be a function such that $f(x+y)=f(x)+f(y), \forall x, y \in R$. If $f(x)$ is differentiable at $x=0$, then

JEE Advanced 2011 Hard

Consider the following lists:

	List- $I$		List- $II$
$(I)$	$\{x \in [- \frac{2 π}{3}, \frac{2 π}{3}] : \cos x + \sin x = 1\}$	$(P)$	has two elements
$(II)$	$\{x \in [- \frac{5 π}{18}, \frac{5 π}{18}] : \sqrt{3} \tan 3 x = 1\}$	$(Q)$	has three elements
$(III)$	$\{x \in [- \frac{6 π}{5}, \frac{6 π}{5}] : 2 \cos (2 x) = \sqrt{3}\}$	$(R)$	has four elements
$(IV)$	$\{x \in [- \frac{7 π}{4}, \frac{7 π}{4}] : \sin x - \cos x = 1\}$	$(S)$	has five elements
		$(T)$	has six elements

The correct option is:

JEE Advanced 2022 Medium

Explore more questions on app

From JEE Advanced

Explore more questions on app