Question

Find \(\int \,3 \,cos⁡x+\frac{1}{x} dx\).

(a) \(3 \,sin⁡x-\frac{1}{x}+C\)

(b) \(2 \,sin⁡x+\frac{1}{x^3}+C\)

(c) \(3 \,sin⁡3x+\frac{1}{x}+C\)

(d) \(sin⁡x-\frac{1}{x^2}+C\)

I have been asked this question in an online quiz.

My enquiry is from Integration as an Inverse Process of Differentiation topic in portion Integrals of Mathematics – Class 12

Answer 1

Right choice is (a) \(3 \,sin⁡x-\frac{1}{x}+C\)

The best I can explain: To find \(\int \,3 \,cos⁡x+\frac{1}{x^2} dx\)

\(\int \,3 \,cos⁡x+\frac{1}{x^2} dx=3 \int cos⁡x \,dx+\int \frac{1}{x^2} \,dx\)

\(\int \,3 \,cos⁡x+\frac{1}{x^2} dx=3 \,sin⁡x+\int x^{-2} \,dx\)

\(\int \,3 \,cos⁡x+\frac{1}{x^2} dx=3 \,sin⁡x+\frac{x^{-2+1}}{-2+1}\)

\(\int \,3 \,cos⁡x+\frac{1}{x^2} dx=3 \,sin⁡x-\frac{1}{x}+C\)