Integration Basics

Reverse the power rule to integrate polynomials (ignore +C in drills).

Difficulty: Easy → Hard Keywords: integration, antiderivative, power rule, SPM Add Maths
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Overview

Integration is the reverse of differentiation. For polynomials, the power rule is simple: to integrate x^n, increase the power by 1 and divide by the new power. For example, \( \int x^2 dx = x^3/3 \). This topic emphasizes accuracy with coefficients and signs, and it builds the habit of writing clean intermediate steps. The drills here follow the common exam format: integrate polynomials quickly, ignore the constant of integration when instructed, and present answers in a tidy form. Once you master the basics, you can handle areas under curves and more advanced integration techniques with much more confidence. A quick self-check is to differentiate your answer mentally; if you get the original expression, your integration is correct. Work through the examples, then try a timed set to strengthen speed and precision.

Worked Examples

Integrate: $$ \int 4x^2\,dx $$
  1. Increase power: x^3, divide by 3.
  2. Result: $$ \frac{4}{3}x^3 $$
$$ \frac{4}{3}x^3 $$
Integrate: $$ \int (5x^3-2x)\,dx $$
  1. Integrate term-by-term.
  2. Result: $$ \frac{5}{4}x^4 - x^2 $$
$$ \frac{5}{4}x^4 - x^2 $$
Integrate: $$ \int (8x^3-4x+9)\,dx $$
  1. Apply the power rule to each term.
  2. Result: $$ 2x^4 - 2x^2 + 9x $$
$$ 2x^4 - 2x^2 + 9x $$

FAQ

Do I always add +C?

In formal integration, yes. In these drills you may be asked to ignore +C.

What if the coefficient is a fraction?

Keep it exact. Divide by the new power and simplify if possible.

Why does integration feel slower than differentiation?

Because you must be careful with coefficients and division. Practice makes it fast.