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Antiderivatives

Reversing differentiation: finding a function whose derivative is a given expression.

Going Backwards

Because the derivative of any constant is 0, x^{3}, x^{3} + 5, and x^{3} - 100 all have the exact same derivative, 3 x^{2} . So an antiderivative is never one single answer — it is a whole family, written with + C to stand for "any constant." Rule (reverse of the power rule): the antiderivative of x^{n} is x^{n + 1} \div (n + 1) + C, as long as n \neq - 1 .

Example 1 — antiderivative of x²

Find the antiderivative of x^{2} . 1 Increase the power by 1 : x^{2} becomes x^{3} 2 Divide by the new power: x^{3} \div 3 3 Add the constant: + C Answer: \frac{x ^{3}}{3} + C . Check: the derivative of \frac{x ^{3}}{3} is \frac{1}{3} \times 3 x^{2} = x^{2} . ✓

Example 2 — antiderivative of x⁴

Find the antiderivative of x^{4} . 1 Increase the power: x^{4} becomes x^{5} 2 Divide by the new power: x^{5} \div 5 3 Add + C Answer: \frac{x ^{5}}{5} + C

Practice

What is the antiderivative of x^{3} ?

Going Backwards

Because the derivative of any constant is 0, x^{3}, x^{3} + 5, and x^{3} - 100 all have the exact same derivative, 3 x^{2} . So an antiderivative is never one single answer — it is a whole family, written with + C to stand for "any constant." Rule (reverse of the power rule): the antiderivative of x^{n} is x^{n + 1} \div (n + 1) + C, as long as n \neq - 1 .

Example 1 — antiderivative of x²

Find the antiderivative of x^{2} . 1 Increase the power by 1 : x^{2} becomes x^{3} 2 Divide by the new power: x^{3} \div 3 3 Add the constant: + C Answer: \frac{x ^{3}}{3} + C . Check: the derivative of \frac{x ^{3}}{3} is \frac{1}{3} \times 3 x^{2} = x^{2} . ✓

Example 2 — antiderivative of x⁴

Find the antiderivative of x^{4} . 1 Increase the power: x^{4} becomes x^{5} 2 Divide by the new power: x^{5} \div 5 3 Add + C Answer: \frac{x ^{5}}{5} + C

Practice

What is the antiderivative of x^{3} ?