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Maharashtra State BoardHSC Science (General) 12th Standard Board Exam

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∫xcos3x dx - Mathematics and Statistics

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Question

$\int x \cos^{3} x d x$

Sum

Solution

Let I = $\int x \cos^{3} x d x$

cos³x = 4cos³x − 3cosx

∴ 4cos³x = 3cos x + cos 3x

∴ cos³x = $\frac{1}{4} (3 \cos x + \cos 3 x)$

∴ I = $\frac{1}{4} \int x (3 \cos x + \cos 3 x) d x$

= $\frac{1}{4} [x \int (3 \cos x + \cos 3 x) d x - \int {\frac{d}{d x} (x) \int (3 \cos x + \cos 3 x) d x} d x]$

= $\frac{1}{4} [x (3 \sin x + \frac{\sin 3 x}{3}) - \int 1 (3 \sin x + \frac{\sin 3 x}{3}) d x]$

= $\frac{1}{4} [3 x \sin x + \frac{x}{3} \sin 3 x - (- 3 \cos x - \frac{1}{3} \cdot \frac{\cos 3 x}{3})] + c$

∴ I = $\frac{1}{4} (3 x \sin x + \frac{x}{3} \sin 3 x + 3 \cos x + \frac{1}{9} \cos 3 x) + c$

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Methods of Integration: Integration Using Partial Fractions

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Chapter 2.3: Indefinite Integration - Long Answers III

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SCERT Maharashtra Mathematics and Statistics (Arts and Science) [English] 12 Standard HSC

Chapter 2.3 Indefinite Integration
Long Answers III | Q 12

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