Integration

Calculus Level 1

Which answer choice is equal to ∫ sin ⁡ x d x x \displaystyle \int \dfrac{\sin \sqrt{x}\text{ d}x }{\sqrt{x}} ?

2 cos ⁡ x + C 2 \cos \sqrt{x}+C − 2 cos ⁡ x + C -2\cos\sqrt{x}+C 4 cos ⁡ x + C 4\cos\sqrt{x}+C cos ⁡ x + C \cos \sqrt{x}+C

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Abhishek Sannake by Abhishek Sannake , 24, India

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1 solution

Caleb Townsend
Apr 7, 2015

... + C u = x , d u = d x 2 x ∫ sin ⁡ u x d x = 2 ∫ x sin ⁡ u x d u = 2 ∫ sin ⁡ u d u = − 2 cos ⁡ u + C = − 2 cos ⁡ x + C u = \sqrt{x},\ \ du = \frac{dx}{2\sqrt{x}} \\ \int \frac{\sin u}{\sqrt{x}} dx = 2\int \frac{\sqrt{x} \sin u}{\sqrt{x}} du \\ = 2\int \sin u\ du \\ = -2 \cos u + C \\ = \boxed{-2\cos\sqrt{x} + C}

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