Integrate sin^3⁡(x+2).

Category: QuestionsIntegrate sin^3⁡(x+2).
Editor">Editor Staff asked 11 months ago

Integrate sin^3⁡(x+2).
(a) \frac{3}{4} \,(sin⁡(x+2))+\frac{1}{12} \,cos⁡(3x+6)+C
(b) –\frac{3}{4} \,(cos⁡(x+2))-\frac{1}{5} \,cos⁡(3x+6)+C
(c) –\frac{3}{4} \,(cos⁡(x+2))+\frac{1}{12} \,cos⁡(3x+6)+C
(d) –\frac{3}{4} \,(cos⁡(x+2))+\frac{1}{12} \,sin⁡(x+2)+C
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Enquiry is from Methods of Integration-2 topic in chapter Integrals of Mathematics – Class 12
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1 Answers
Editor">Editor Staff answered 11 months ago

Right option is (c) –\frac{3}{4} \,(cos⁡(x+2))+\frac{1}{12} \,cos⁡(3x+6)+C
Explanation: To find: ∫ 3 sin^3⁡(x+2) dx
We know that, sin⁡3x=3 sin⁡x-4 sin^3⁡x
∴sin^3⁡⁡x=\frac{3 sin⁡x-sin⁡3x}{4}
sin^3⁡(x+2)=\frac{(3 sin⁡(x+2)-sin⁡(3x+6))}{4}
\int sin^3⁡(x+2) \,dx=\frac{3}{4} \int sin⁡(x+2) \,dx-\frac{1}{4} \int \,sin⁡(3x+6) \,dx
=-\frac{3}{4} \,(cos⁡(x+2))+\frac{1}{12} \,cos⁡(3x+6)+C