Evaluate - -2 sin 3x- sin 6 xdx

sin (a) ×sin (b) = 1 2 ×[ cos( a b ) #160; cos( a+b ) #160; #160; ] ∴ 2sin( 3x ) #160; ×sin( 6x ) #160; =2× 1 2 ×[ cos( 3x 6x ) #16 ...

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Byju's Answer

Standard XII

Mathematics

Integration of Trigonometric Functions

Evaluate : ...

Question

Evaluate : $\int 2 sin 3 x . sin 6 x d x$

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Solution

$sin (a) \times sin (b) = \frac{1}{2} \times [cos (a - b) - cos (a + b)] ∴ 2 sin (3 x) \times sin (6 x) = 2 \times \frac{1}{2} \times [cos (3 x - 6 x) - cos (3 x + 6 x)] => 2 sin (3 x) \times sin (6 x) = cos (3 x) - cos (9 x) ∴ I = \int 2 sin (3 x) \times sin (6 x) d x = \int (cos (3 x) - cos (9 x)) d x = \frac{- 1}{3} sin (3 x) + \frac{1}{9} sin (9 x) + c [∵ \int cos (a x) d x = \frac{- 1}{a} sin (a x) + c]$

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Evaluate : ∫2sin3x.sin6xdx

sin(a)×sin(b)=12×[cos(a−b)−cos(a+b)]∴2sin(3x)×sin(6x)=2×12×[cos(3x−6x)−cos(3x+6x)]=>2sin(3x)×sin(6x)=cos(3x)−cos(9x)∴I=∫2sin(3x)×sin(6x)dx=∫(cos(3x)−cos(9x))dx=−13sin(3x)+19sin(9x)+c[∵∫cos(ax)dx=−1asin(ax)+c]

Evaluate : $\int 2 sin 3 x . sin 6 x d x$