Finding antiderivatives

An antiderivative F is a function whose derivative is f. Integrating is differentiating backwards, and the check is always the same: differentiate F.

Rules

Three problems on this topic. Type like on a calculator: 3x^2 - 2/x, e^(2x), sqrt(x), ln(x).

Task 1 of 3Antiderivatives

Task

Find an antiderivative:

f (x) = 12 x^{2} + 2 x + 6

$F (x) =$ (with or without $+ C$ )

Write like on a calculator: 3x^2 - 2/x, e^(2x), sqrt(x), ln(x), sin(x). Use brackets: e^(2x), not e^2x. Numbers also as 3/4 or e^2 - 1.

Worked example

Find an antiderivative: $f (x) = 12 x^{2} + 8 x - 3$

Worked solution

1.Power rule backwards: $\int x^{n} d x = \frac{x ^{n + 1}}{n + 1} + C$ (for $n \neq = - 1$ ).
2.Integrate each term; a constant $a$ becomes $a x$ .
3. $F (x) = 4 x^{3} + 4 x^{2} - 3 x + C$
4.Check: $F^{'} (x) = f (x)$ .

Typical mistake: Dividing by the old instead of the new exponent: $\int x^{2} d x = \frac{x ^{3}}{3}$ , not $\frac{x ^{3}}{2}$ .

Find an antiderivative: $f (x) = 4 e^{x} - \frac{6}{x} - 2 cos (x)$

Worked solution

1.Basic integrals: $\int e^{x} d x = e^{x}$ , $\int \frac{1}{x} d x = ln ∣ x ∣$ , $\int cos x d x = sin x$ , $\int sin x d x = - cos x$ .
2. $F (x) = 4 e^{x} - 6 ln ∣ x ∣ - 2 sin (x) + C$

Typical mistake: Using the power rule on $\int \frac{1}{x} d x$ : $\frac{x ^{0}}{0}$ is undefined, the antiderivative is $ln ∣ x ∣$ .

Common questions

Every constant disappears when you differentiate. So every function has infinitely many antiderivatives that differ only by a constant C.

Differentiate it. If you get the original function back, it is correct.