Evaluate the integral ∫ (3x^2+2x) dx and choose the correct antiderivative.

Evaluate the integral $\int (3 x^{2} + 2 x) d x$ and choose the correct antiderivative.

easy MCQ 1 mark Class 12 · Mathematics · Integrals

Answer

x^{3} + x^{2} + C

The correct answer is $x^{3} + x^{2} + C$ — Add the two results and include the constant of integration $C$ to obtain $x^{3} + x^{2} + C$ .

Identify the integrand as a sum of two power terms, $3 x^{2}$ and $2 x$ , because integration is linear.
Integrate $3 x^{2}$ using the power rule $\int x^{n} d x = \frac{x ^{n + 1}}{n + 1}$ giving $3 \cdot \frac{x ^{3}}{3} = x^{3}$ .
Integrate $2 x$ similarly to obtain $2 \cdot \frac{x ^{2}}{2} = x^{2}$ .
Add the two results and include the constant of integration $C$ to obtain $x^{3} + x^{2} + C$ .

Common mistake: Forgetting the constant of integration

C

after finding the antiderivative.

Concept tested: basic integration