Antiderivatives

Warm-up

Three quick questions from earlier lessons.

Discover

Which function has derivative $3x^2$ ?

$x^3$
$6x$

Key idea

Run the derivative rules backwards. Antiderivatives differ by a constant $C$ .

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

(

n \ne -1

);

\displaystyle\int \dfrac{1}{x}\,dx = \ln|x| + C

;

\displaystyle\int e^x\,dx = e^x + C

;

\displaystyle\int \cos x\,dx = \sin x + C

;

\displaystyle\int \sin x\,dx = -\cos x + C

. A given value such as

F(0)

fixes

C

Worked example

Find $F$ with $F'(x) = 4x^3 + \sin x$ and $F(0) = 2$ .

F(x) = x^4 - \cos x + C

Term by term.

C = 3

F(0) = 0 − 1 + C = 2.

F(x) = x^4 - \cos x + 3

Check by differentiating.

Your turn, guided

Your turn: $F'(x) = 6x^2$ , $F(0) = 0$ .

F(x) = 2x^3

6x³/3.

Practice ★

Find the antiderivative $F$ of $f(x) = 4x^{ 3 } + 9$ with $F(0) = 0$ .
Find the antiderivative $F$ of $f(x) = 2x^{ 2 } + 8$ with $F(0) = 0$ .
Find the antiderivative $F$ of $f(x) = 5x^{ 4 } + 6$ with $F(0) = 0$ .

Watch out!

Practice ★★

Find the antiderivative $F$ of $f(x) = \cos x + e^x$ with $F(0) = 6$ .
Find the antiderivative $F$ of $f(x) = \cos x + e^x$ with $F(0) = 4$ .

Find the antiderivative $F$ of $f(x) = \dfrac{1}{x}$ , for $x > 0$ , with $F(1) = -9$ .
Find the antiderivative $F$ of $f(x) = \dfrac{1}{x}$ , for $x > 0$ , with $F(1) = 8$ .

Exit ticket

Three questions to check you have got it.

Find the antiderivative $F$ of $f(x) = 4x^{ 3 } + 9$ with $F(0) = 0$ .
Find the antiderivative $F$ of $f(x) = 2x^{ 2 } + 8$ with $F(0) = 0$ .
Find the antiderivative $F$ of $f(x) = 5x^{ 4 } + 6$ with $F(0) = 0$ .

Summary card

Antiderivatives

$x^n \to \dfrac{x^{n+1}}{n+1}$ ; $\tfrac{1}{x} \to \ln|x|$ ; $e^x \to e^x$ ; $\cos \to \sin$ ; $\sin \to -\cos$ .
Use a given value to find $C$ .

Watch out: check by differentiating.

End of the lesson

Next lesson: The Fundamental Theorem of Calculus: evaluating integrals Back to the chapter

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