Question

Find ∫ 7x^2-x^3+2x dx.

(a) \(\frac{7x^3}{3}+\frac{x^4}{5}-\frac{2x^2}{2}+C\)

(b) \(\frac{7x^3}{3}+\frac{x^4}{4}+\frac{2x^2}{2}+C\)

(c) \(\frac{7x^5}{9}-\frac{x^4}{4}+\frac{2x^2}{2}+C\)

(d) \(\frac{7x^3}{3}-\frac{x^4}{4}+x^2+C\)

I have been asked this question by my school principal while I was bunking the class.

I'd like to ask this question from Integration as an Inverse Process of Differentiation in division Integrals of Mathematics – Class 12

Answer 1

The correct option is (d) \(\frac{7x^3}{3}-\frac{x^4}{4}+x^2+C\)

Explanation: To find \(\int 7x^2-x^3+2x dx\)

\(\int 7x^2-x^3+2x dx=\int 7x^2 dx-\int x^3 dx+2\int x dx\)

Using \(\int x^n dx=\frac{x^{n+1}}{n+1}\), we get

\(\int 7x^2-x^3+2x dx=\frac{7x^{2+1}}{2+1}-\frac{x^{3+1}}{3+1}+2(\frac{x^{1+1}}{1+1})\)

∴\(\int 7x^2-x^3+2x dx=\frac{7x^3}{3}-\frac{x^4}{4}+x^2+C\)