Question

Find the integral of \(\int 3e^x+\frac{2}{x}+x^3 dx\).

(a) \(3e^3x+\frac{2}{x}-\frac{x^4}{4}+c\)

(b) \(3e^x+2 \,log⁡x+\frac{x^4}{4}+c\)

(c) \(e^x+2 \,log⁡x+\frac{x^4}{4}+c\)

(d) \(3e^x-\frac{2}{x^2}+\frac{x^4}{4}+c\)

I got this question in a job interview.

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

Answer 1

The correct answer is (b) \(3e^x+2 \,log⁡x+\frac{x^4}{4}+c\)

The explanation: To find \(\int \,3e^x+\frac{2}{x}+x^3 \,dx\)

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

\(\int \,e^x \,dx=e^x\)

\(\int \frac{1}{x} dx=log⁡x\)

∴\(\int 3e^x+\frac{2}{x}+x^3 \,dx=3e^x+2 \,log⁡x+\frac{x^4}{4}+c\)