If ∫12dxx2-2x+4^32=k/(k+5), then k is equal to

Correct answer: (D) 1. ∫ 1 2 dx/(( ( x−1 )^2 +3 ) 3 2) Substituting x-1=√3tan⁡θ ⇒dx=√3sec^2θ·dθ Now ∫0(π)/6(√3 sec^2 θ ⁡dθ)/(3√3 sec^3 θ) =1/3∫0(π)/6cosθdθ=1/3sin⁡θ0(π)/6=1/6 ⇒1/6=k/(k+5)=k+5=6k =k=1.

Q35 / 90 · JEE Main 2017 (09 Apr Online) · Mathematics

If $\int_{1}^{2} \frac{d x}{{(x^{2} - 2 x + 4)}^{\frac{3}{2}}} = \frac{k}{k + 5}$ , then $k$ is equal to

A
$4$
B
$2$
C
$3$
Correct answer
D
$1$

Solution

\int_{1}^{2} \frac{d x}{{({(x - 1)}^{2} + 3)}^{\frac{3}{2}}}

Substituting

x - 1 = \sqrt{3} \tan θ ​

\Rightarrow d x = \sqrt{3} \sec^{2} θ \cdot d θ ​

Now

\int_{0}^{\frac{π}{6}} \frac{\sqrt{3} \sec^{2} θ​ d θ​}{3 \sqrt{3} \sec^{3} θ​}

= \frac{1}{3} \int_{0}^{\frac{π}{6}} \cos θ d θ ​ = \frac{1}{3} {[\sin θ]}_{0}^{\frac{π}{6}} = \frac{1}{6}

\Rightarrow \frac{1}{6} = \frac{k}{k + 5} = k + 5 = 6 k

= k = 1

Hence, the correct answer is option: 'D'.

If ∫12dxx2-2x+432=kk+5, then k is equal to

Solution

Relevant Questions

If $\int_{1}^{2} \frac{d x}{{(x^{2} - 2 x + 4)}^{\frac{3}{2}}} = \frac{k}{k + 5}$ , then $k$ is equal to