Visual SolutionPYQ 2024 · Jan Shift 2Standard

Question

\int_0^{\pi/3} \cos^4 x \, dx = a\pi + b\sqrt{3}

, where

a

and

b

are rational numbers, then

9a + 8b

is equal to:

(A)

2

(B)

1

(C)

3

(D)

\frac{3}{2}

Solution Path

Power reduction:

\cos^4 x = \frac{3}{8} + \frac{\cos 2x}{2} + \frac{\cos 4x}{8}

. Integrate to get

a=\frac{1}{8}, b=\frac{7}{64}

. Answer:

9a+8b=2

01Question Setup

1/4

\int_0^{\pi/3} \cos^4 x\, dx = a\pi + b\sqrt{3}

. Find

9a + 8b

9a + 8b = \;?

02Power Reduction

2/4

\cos^4 x = \frac{3}{8} + \frac{\cos 2x}{2} + \frac{\cos 4x}{8}

using double angle identities.

\cos^4 x = \frac{3}{8} + \cdots

03EvaluateKEY INSIGHT

3/4

Integrating:

\frac{\pi}{8} + \frac{7\sqrt{3}}{64}

. So

a = \frac{1}{8}, b = \frac{7}{64}

. Then

9a + 8b = 2

9a + 8b = 2

04Final Answer

4/4

9a + 8b = 2

\boxed{2}

Concepts from this question2 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

EASY

For symmetric limits, odd functions integrate to zero and even functions double

\int_{-a}^{a} f(x)\,dx = \begin{cases} 0 & f(-x) = -f(x) \\ 2\int_0^a f(x)\,dx & f(-x) = f(x) \end{cases}

Saves computation by detecting when an integral vanishes or can be simplified

Definite integralsTrigonometric integrals

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