Find the antiderivative of the function $4 x^{3} - 5 x^{2} + 2$ and write the general form of the integral $\int (4 x^{3} - 5 x^{2} + 2), d x$ .

easy 1 mark Class 12 · Mathematics · Integrals

Answer

x^{4} - \frac{5}{3} x^{3} + 2 x + C

The correct answer is $x^{4} - \frac{5}{3} x^{3} + 2 x + C$ — Add the constant of integration $C$ because the antiderivative is indefinite.

Step-by-step solution

Apply the power rule to each term separately. The integral of $4 x^{3}$ is $4 \cdot \frac{x ^{4}}{4} = x^{4}$ .
The integral of $- 5 x^{2}$ is $- 5 \cdot \frac{x ^{3}}{3} = - \frac{5}{3} x^{3}$ .
The integral of the constant $2$ is $2 x$ because $in t 2, d x = 2 x$ .
Add the constant of integration $C$ because the antiderivative is indefinite.

Common mistake: Forgetting to add the constant of integration

Concept tested: Basic antiderivative

Published 5 September 2026 · Last updated 16 September 2026 · How this solution was prepared and checked