Question

9. Evaluate (a) \( \int \sin 3 x \cos 5 x d x \) (b) \( \int \cos \frac{x}{2} \cos \frac{x}{3} d x \) (c) \( \int \sin x \sin 2 x \sin 3 x d x \) (Ans: (a) \( \frac{1}{4} \cos 2 x-\frac{1}{16} \cos 8 x+C \), (b) \( \frac{3}{5} \sin \frac{5}{6} x+3 \sin \frac{1}{6} x+C \), (c) \( \frac{1}{24} \cos 6 x-\frac{1}{16} \cos 4 x- \) \( \frac{1}{8} \cos 2 x+C \) )

          9. Evaluate
(a) \( \int \sin 3 x \cos 5 x d x \)
(b) \( \int \cos \frac{x}{2} \cos \frac{x}{3} d x \)
(c) \( \int \sin x \sin 2 x \sin 3 x d x \)
(Ans: (a) \( \frac{1}{4} \cos 2 x-\frac{1}{16} \cos 8 x+C \),
(b) \( \frac{3}{5} \sin \frac{5}{6} x+3 \sin \frac{1}{6} x+C \),
(c) \( \frac{1}{24} \cos 6 x-\frac{1}{16} \cos 4 x- \) \( \frac{1}{8} \cos 2 x+C \) )

9. Evaluate
(a) ∫sin 3 x cos 5 x d x
(b) ∫cos(x)/(2)cos(x)/(3) d x
(c) ∫sin x sin 2 x sin 3 x d x
(Ans: (a) (1)/(4)cos 2 x-(1)/(16)cos 8 x+C,
(b) (3)/(5)sin(5)/(6) x+3 sin(1)/(6) x+C,
(c) (1)/(24)cos 6 x-(1)/(16)cos 4 x- (1)/(8)cos 2 x+C )

Added by Dustin W.

Calculus: Early Transcendentals

James Stewart 8th Edition

Instant Answer

Solved by Expert Madhur L

Step 1

- Use the product-to-sum identities: \[ \sin A \cos B = \frac{1}{2} [\sin(A+B) + \sin(A-B)] \] - Apply the identity: \[ \sin 3x \cos 5x = \frac{1}{2} [\sin(8x) + \sin(-2x)] \] - Integrate: \[ \int \sin 3x \cos 5x \, dx = \frac{1}{2} \left( \int Show more…

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9. Evaluate (a) \( \int \sin 3 x \cos 5 x d x \) (b) \( \int \cos \frac{x}{2} \cos \frac{x}{3} d x \) (c) \( \int \sin x \sin 2 x \sin 3 x d x \) (Ans: (a) \( \frac{1}{4} \cos 2 x-\frac{1}{16} \cos 8 x+C \), (b) \( \frac{3}{5} \sin \frac{5}{6} x+3 \sin \frac{1}{6} x+C \), (c) \( \frac{1}{24} \cos 6 x-\frac{1}{16} \cos 4 x- \) \( \frac{1}{8} \cos 2 x+C \) )