Apply the power rule of integration to find ∫x²dx step by step.

Integrate ∫x²dx — Basic Power Rule

Question

Find the integral:

\int x^2 \, dx

This is a foundational problem from NCERT Class 12, Chapter 7. Master this and the entire power rule family becomes mechanical.

Solution — Step by Step

The power rule states: for any $n \neq -1$ ,

\int x^n \, dx = \frac{x^{n+1}}{n+1} + C

Here, $n = 2$ . We increase the power by 1 and divide by the new power. Simple as that.

Substitute $n = 2$ directly:

\int x^2 \, dx = \frac{x^{2+1}}{2+1} + C

= \frac{x^3}{3} + C

That’s it. The constant of integration $C$ is mandatory — dropping it is a board exam error.

Final Answer:

\boxed{\int x^2 \, dx = \frac{x^3}{3} + C}

Why This Works

Integration is the reverse of differentiation. When we differentiate $\frac{x^3}{3}$ , we bring the power down as a coefficient and reduce the exponent by 1:

\frac{d}{dx}\left(\frac{x^3}{3}\right) = \frac{3x^2}{3} = x^2

So integrating $x^2$ simply reverses that process — we go from exponent 2 back to exponent 3, and divide by 3 to cancel what differentiation would multiply.

The $+ C$ appears because any constant vanishes on differentiation. When we reverse the process, we can’t know which constant was there originally — so we write the general form with $C$ .

Alternative Method — Verification by Differentiation

Always verify your integral by differentiating the result. If you get back the integrand, you’re correct.

\frac{d}{dx}\left(\frac{x^3}{3} + C\right) = \frac{1}{3} \cdot 3x^2 + 0 = x^2 \checkmark

This verification habit is worth building now. In JEE, when you’re unsure about a complex integral, a quick differentiation check can save marks.

Power rule works for all rational and real exponents — not just positive integers. For example, $\int x^{1/2} \, dx = \frac{x^{3/2}}{3/2} + C = \frac{2}{3}x^{3/2} + C$ . Same formula, same logic.

Common Mistake

Forgetting to divide by the new power. Many students write $\int x^2 \, dx = x^3 + C$ , correctly incrementing the exponent but skipping the division step. The power rule has two operations: raise the power AND divide by it. Forgetting the division is a guaranteed -1 in boards and a wrong option in JEE MCQs.

A useful self-check: differentiate your answer mentally. If you get back $x^2$ cleanly, you’re right. If you get $3x^2$ , you forgot to divide.

Question

Solution — Step by Step

Why This Works

Alternative Method — Verification by Differentiation

Common Mistake

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