Integrate 5x sin⁡3x.

Category: QuestionsIntegrate 5x sin⁡3x.
Editor">Editor Staff asked 11 months ago

Integrate 5x sin⁡3x.
 
(a) –\frac{5}{3} \,x \,cos⁡3x+\frac{5}{9} tan⁡3x+C
 
(b) \frac{5}{3} \,cos⁡3x-\frac{5}{9} \,sin⁡3x+C
 
(c) x cos⁡3x+\frac{5}{9} \,sin⁡3x+C
 
(d) –\frac{5}{3} \,x \,cos⁡3x+\frac{5}{9} \,sin⁡3x+C
 
I got this question in an internship interview.
 
The question is from Integration by Parts in portion 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 answer is (d) –\frac{5}{3} \,x \,cos⁡3x+\frac{5}{9} \,sin⁡3x+C
 
Easy explanation: By using the formula, ∫ u.v dx=u∫ v dx-∫ u’ (∫ v dx) ,we get
 
∫ 5x sin⁡3x dx=5x∫ sin⁡3x dx -∫ (5x)’∫ sin⁡3x dx
 
=5x(-\frac{cos⁡3x}{3})+\int \frac{5 cos⁡3x}{3} dx
 
=-\frac{5}{3} x cos⁡3x+\frac{5}{9} sin⁡3x+C