Ex 7.6, 21 - Chapter 7 Class 12 Integrals

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Ex 7.6, 21 - Chapter 7 Class 12 Integrals - NCERT Solution Integrate e^2x sin x I = ∫ e^2x sin x dx Using ILATE e^2x -> Exponential sin x -> Trigonometric We know that ∫ f(x) g(x) dx = f(x) ∫ g(x) dx - ∫ (f'(x) ∫ g(x)dx)dx Putting f(x) = e^2x, g(x) = sin x I = sin . 2 I = sin 2 sin 2 I = sin . 2 2 cos . 2 2 I = 1 2 . 2 sin 1 2 cos . 2 Solving I2 = 1 2 cos . 2 I1 = 1 2 cos . 2 = 1 2 cos 2 cos 2 = 1 2 cos . 2 2 ( sin ) . 2 2 = 1 2 2 . cos 2 + 1 2 2 sin = 1 2 2 . cos 2 + 1 2 I + 1 Putting the value of I1 in (1) , we get I = 2 sin I = 2 sin 2 1 2 2 . cos 2 + I 2 + 1 I = 2 2 sin 2 4 cos I 4 1 + I 4 = 2 sin 2 2 . cos 4 5 4 = 2 4 2 sin cos 1 = 4 5 . 2 4 2 sin cos 4 1 5 = +