20 min read

Integration Rules

The indefinite integral is another name for the antiderivative, with the same sum, difference, and constant multiple rules.

Integral Notation

Example 1 — integrating a sum

Find \int(3 x^{2} + 2 x)dx. 1 Integrate each term: \int 3 x^{2} dx = 3 \times \frac{x ^{3}}{3} = x^{3} . \int 2 x dx = 2 \times \frac{x ^{2}}{2} = x^{2} 2 Add the results, and add + C at the end Answer: x^{3} + x^{2} + C

Example 2 — integrating a difference

Find \int(4 x^{3} - 6 x)dx. 1 \int 4 x^{3} dx = 4 \times \frac{x ^{4}}{4} = x^{4} 2 \int 6 x dx = 6 \times \frac{x ^{2}}{2} = 3 x^{2} 3 Subtract, and add + C Answer: x^{4} - 3 x^{2} + C

Practice

What is \int 6 x^{2} dx?

20 min read

The indefinite integral is another name for the antiderivative, with the same sum, difference, and constant multiple rules.

Example 1 — integrating a sum

Find \int(3 x^{2} + 2 x)dx. 1 Integrate each term: \int 3 x^{2} dx = 3 \times \frac{x ^{3}}{3} = x^{3} . \int 2 x dx = 2 \times \frac{x ^{2}}{2} = x^{2} 2 Add the results, and add + C at the end Answer: x^{3} + x^{2} + C

Example 2 — integrating a difference

Find \int(4 x^{3} - 6 x)dx. 1 \int 4 x^{3} dx = 4 \times \frac{x ^{4}}{4} = x^{4} 2 \int 6 x dx = 6 \times \frac{x ^{2}}{2} = 3 x^{2} 3 Subtract, and add + C Answer: x^{4} - 3 x^{2} + C

Practice

What is \int 6 x^{2} dx?