Visual SolutionPYQ 2024 · Apr Shift 1Tricky

Question

Let

I(x) = \int \frac{6}{\sin^2 x(1-\cot x)^2} \, dx

. If

I(0) = 3

, then

I\left(\frac{\pi}{12}\right)

is equal to:

(A)

2\sqrt{3}

(B)

\sqrt{3}

(C)

3\sqrt{3}

(D)

6\sqrt{3}

Solution Path

Sub

t = 1-\cot x

gives

I = -6/(1-\cot x) + 3

. At

x = \pi/12

\cot = 2+\sqrt{3}

, so

I = 3\sqrt{3}

01Question Setup

1/4

I(x) = \int \frac{6}{\sin^2 x\,(1-\cot x)^2}\,dx

with

I(0) = 3

. Find

I(\frac{\pi}{12})

I(\frac{\pi}{12}) = \;?

02Substitution

2/4

Let

t = 1 - \cot x

dt = \csc^2 x\,dx

. Integral becomes

-\frac{6}{1-\cot x} + C

I = \frac{-6}{1-\cot x} + C

03Apply ConditionsKEY INSIGHT

3/4

I(0) = 3 \Rightarrow C = 3

. With

\cot\frac{\pi}{12} = 2+\sqrt{3}

I = \frac{6}{1+\sqrt{3}} + 3 = 3\sqrt{3}

3\sqrt{3}

04Final Answer

4/4

I(\frac{\pi}{12}) = 3\sqrt{3}

\boxed{3\sqrt{3}}

Concepts from this question1 concepts unlocked

★

EASY

Replace a composite expression with a single variable to simplify the integrand

\int f(g(x))\,g'(x)\,dx = \int f(t)\,dt, \quad t = g(x)

Transforms complex integrands into standard forms that can be integrated directly

Trigonometric integralsRational functionsPower rule integrals

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