LemonmathSelf-study sheet

Grade 12 · Lesson 4.1

Antiderivatives

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Goals

Understand an antiderivative of a function and the family $F(x)+C$ .
Use the table of standard antiderivatives and the two linearity rules to find $\displaystyle\int f(x)\,dx$ .
Find the antiderivative that meets a given condition, and check the result by differentiating.
Find the distance travelled from the velocity in a motion problem.

Remember

$F$ is an antiderivative of $f$ on $K$ when $F'(x)=f(x)$ for every $x\in K$ .
Every antiderivative has the form $F(x)+C$ ; we write $\displaystyle\int f(x)\,dx=F(x)+C$ .
The table to remember: $x^\alpha\to\dfrac{x^{\alpha+1}}{\alpha+1}$ ( $\alpha\neq-1$ ), $\dfrac1x\to\ln\lvert x\rvert$ , $e^x\to e^x$ , $a^x\to\dfrac{a^x}{\ln a}$ , $\cos x\to\sin x$ , $\sin x\to-\cos x$ , $\dfrac1{\cos^2x}\to\tan x$ , $\dfrac1{\sin^2x}\to-\cot x$ .
Constant factors, sums and differences split apart; products do not.
Given $F(x_0)=y_0$ , substitute it to find $C$ .
Distance is an antiderivative of velocity; always check by differentiating.

Exercises

Exercise 1

Find $\displaystyle\int(4x^3-6x+7)\,dx$ .

Exercise 2

Find $\displaystyle\int\left(3\sqrt x+\frac1{x^2}\right)dx$ on $(0, +\infty)$ .

Exercise 3

Find $\displaystyle\int\left(5e^x-2\sin x+\frac{1}{\sin^2x}\right)dx$ .

Exercise 4

Find $\displaystyle\int(2x-1)^2\,dx$ .

Exercise 5

Consider $f(x)=3x^2+2\cos x$ . Is each claim true or false?

a) $F(x)=x^3+2\sin x$ is an antiderivative of $f(x)$ .

b) $F(x)=x^3-2\sin x+5$ is an antiderivative of $f(x)$ .

c) The antiderivative $G(x)$ of $f(x)$ with $G(0)=1$ is $G(x)=x^3+2\sin x+1$ .

d) For $G(x)$ in c), $G\left(\dfrac{\pi}{2}\right)=\dfrac{\pi^3}{8}+2$ .

Exercise 6

$F(x)$ is the antiderivative of $f(x)=4x^3-2x$ with $F(1)=3$ . Find $F(2)$ .

Exercise 7

A tank already holds $200$ litres of water. A pump is switched on, and $t$ minutes later the volume of water is changing at the rate $V'(t)=6t+20$ (litres per minute). How many litres are in the tank after $10$ minutes?

Exercise 8 Review

An object moves in a straight line with velocity $v(t)=12t-3t^2$ (m/s), $0\le t\le 4$ . Find its greatest velocity and the distance it travels in the first $4$ seconds.

Exercise 9 Challenge

Find the function $f(x)$ with $f''(x)=6x+2$ , $f'(0)=1$ and $f(0)=-1$ .

Exercise 10 Challenge

Find $\displaystyle\int\sin^2\frac{x}{2}\,dx$ .

Mini test

15 minutes, without looking back at the lesson. Part I: circle one letter; Part II: write T or F for each statement; Part III: write the answer in the box.

Part I. Multiple choice

Question 1. The indefinite integral of

f(x)=3x^2-\sin x

x^3+\cos x+C

.B.

x^3-\cos x+C

.C.

6x-\cos x+C

.D.

3x^3+\cos x+C

Question 2. The indefinite integral of

f(x)=2^x

2^x\ln 2+C

.B.

\dfrac{2^x}{\ln 2}+C

.C.

\dfrac{2^{x+1}}{x+1}+C

.D.

x\cdot 2^{x-1}+C

Question 3.

F(x)

is the antiderivative of

f(x)=2x+1

with

F(1)=5

. The value of

F(2)

6

.B.

11

.C.

9

.D.

5

Question 4. A car brakes with velocity

v(t)=15-3t

(m/s) until it stops. The distance it travels from braking until it stops is

75

m.B.

37.5

m.C.

112.5

m.D.

0

Part II. True or false

Question 1. Consider

f(x)=e^x-2x

a) $F(x)=e^x-x^2$ is an antiderivative of $f(x)$ .	T / F
b) $\displaystyle\int f(x)\,dx=e^x-2+C$ .	T / F
c) The antiderivative $F(x)$ of $f(x)$ with $F(0)=3$ is $F(x)=e^x-x^2+2$ .	T / F
d) For $F(x)$ in c), $F(1)=e+1$ .	T / F

Part III. Short answer

Question 1.

F(x)

is the antiderivative of

f(x)=6x^2-4x+3

with

F(0)=-2

. Find

F(2)

Question 2.

F(x)

is the antiderivative of

f(x)=e^x+\dfrac1x

(0, +\infty)

with

F(1)=e

. Find

F(2)

. Give your answer to 1 decimal place.

Score: ........ / 10

Hints (only when stuck)

Exercise 1.

What you need: $\displaystyle\int x^\alpha dx=\frac{x^{\alpha+1}}{\alpha+1}+C$ and the rule for sums.

Steps:

Find the antiderivative of each term.
Add a constant $C$ and check by differentiating.

Exercise 2.

What you need: write $\sqrt x=x^{\frac12}$ and $\dfrac{1}{x^2}=x^{-2}$ , then use the power formula.

Steps:

Rewrite as powers.
Apply the formula to each term.

Exercise 3.

What you need: the $e^x$ , $\sin x$ and $\dfrac{1}{\sin^2x}$ rows of the table.

Steps:

Write the antiderivative of each term, watching the two minus signs.
Put them together and add $C$ .

Exercise 4.

What you need: do not find the antiderivative of a square by "squaring the antiderivative"; expand first.

Steps:

Expand $(2x-1)^2=4x^2-4x+1$ .
Find the antiderivative of the polynomial.

Exercise 5.

What you need: $F$ is an antiderivative of $f$ when $F'=f$ ; the condition $G(0)=1$ fixes the constant $C$ .

Steps:

Differentiate the functions in a) and b).
Find $C$ in c), then substitute $x=\dfrac{\pi}{2}$ for d).

Exercise 6.

What you need: find the family of antiderivatives, then use $F(1)=3$ to find $C$ .

Steps:

$F(x)=x^4-x^2+C$ .
Substitute $x=1$ to find $C$ , then compute $F(2)$ .

Exercise 7.

What you need: the volume $V(t)$ is an antiderivative of $V'(t)$ ; at the start, $V(0)=200$ .

Steps:

Find $V(t)=3t^2+20t+C$ .
Use $V(0)=200$ to find $C$ , then compute $V(10)$ .

Exercise 8.

What you need: how to find a maximum on a closed interval (see maximum and minimum values); distance is an antiderivative of velocity.

Steps:

Solve $v'(t)=0$ and compare $v(0)$ , $v(2)$ , $v(4)$ .
Find $s(t)$ with $s(0)=0$ , then compute $s(4)$ .

Exercise 9.

What you need: $f'$ is an antiderivative of $f''$ , and $f$ is an antiderivative of $f'$ (second derivatives were met in grade 11).

Steps:

Find $f'(x)$ and use $f'(0)=1$ to find the first constant.
Find $f(x)$ and use $f(0)=-1$ to find the second constant.

Exercise 10.

What you need: the double-angle formula (grade 11): $\cos x=1-2\sin^2\dfrac x2$ , so $\sin^2\dfrac x2=\dfrac{1-\cos x}{2}$ .

Steps:

Replace $\sin^2\dfrac x2$ by $\dfrac12-\dfrac12\cos x$ .
Find the antiderivative term by term.

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