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
 
The question was posed to me during an interview.
 
Enquiry is from Methods of Integration-2 topic in chapter Integrals of Mathematics – Class 12
NCERT Solutions for Subject Clas 12 Math Select the correct answer from above options 
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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