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Amy Ferguson Moncure

Calculus BC ยท Unit 6

Antiderivatives

Build the general antiderivative, then use a starting value when one is given.

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Key Ideas Easy Medium Hard

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

\(n\neq -1\)

What to Remember

Power rule, then add C

For \(n\neq -1\),

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

An indefinite integral is a family of functions.

The four basic transcendental rules

\(\int \dfrac{1}{x}\,dx=\ln|x|+C\), \(\int e^x\,dx=e^x+C\)

\(\int \sin x\,dx=-\cos x+C\), \(\int \cos x\,dx=\sin x+C\)

Two inverse-trig antiderivatives

\(\int \dfrac{1}{1+x^2}\,dx=\arctan x+C\)

\(\int \dfrac{1}{\sqrt{1-x^2}}\,dx=\arcsin x+C\)

A starting value picks one function

Find the general antiderivative, then use \(f(a)=b\) to solve for \(C\).

Level 1

Easy

One rule, plus the constant.

Example
Find \(\int (6x^2-4x+1)\,dx\).

\(\int 6x^2\,dx = 2x^3\)

\(\int -4x\,dx = -2x^2\)

\(\int 1\,dx = x\)

\(2x^3-2x^2+x+C\)

Answer: \(2x^3-2x^2+x+C\)

1
Find \(\int 4x^3\,dx\).
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\(\dfrac{4x^{4}}{4}+C\)

\(x^4+C\)

Answer: \(x^4+C\)

2
Find \(\int 5\,dx\).
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\(5x+C\)

Answer: \(5x+C\)

3
Find \(\int 3e^x\,dx\).
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\(3e^x+C\)

Answer: \(3e^x+C\)

4
Find \(\int \dfrac{4}{x}\,dx\).
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\(4\ln|x|+C\)

Answer: \(4\ln|x|+C\)

5
Find \(\int 2\cos x\,dx\).
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\(2\sin x+C\)

Answer: \(2\sin x+C\)

Level 2

Medium

A negative power, a second basic rule, or one initial condition.

Example
Find \(\int \left(4\sin x+\sec^2 x\right)\,dx\).

\(\int \sin x\,dx = -\cos x\)

\(\int \sec^2 x\,dx = \tan x\)

\(-4\cos x+\tan x+C\)

Answer: \(-4\cos x+\tan x+C\)

1
Find \(\int x^{-3}\,dx\).
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\(\int x^{-3}\,dx = \dfrac{x^{-2}}{-2}+C\)

\(-\dfrac{1}{2x^2}+C\)

Answer: \(-\dfrac{1}{2x^2}+C\)

2
\(f'(x)=6x^2\) and \(f(1)=5\). Find \(f(x)\).
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\(f(x)=2x^3+C\)

\(f(1)=2+C\)

\(2+C=5\)

\(C=3\)

\(f(x)=2x^3+3\)

Answer: \(2x^3+3\)

3
Find \(\int \dfrac{3}{1+x^2}\,dx\).
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\(3\arctan x+C\)

Answer: \(3\arctan x+C\)

4
Find \(\int \dfrac{x^3+2}{x}\,dx\) for \(x>0\).
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\(\dfrac{x^3+2}{x} = x^2+\dfrac{2}{x}\)

\(\int x^2\,dx = \dfrac{x^3}{3}\)

\(\int \dfrac{2}{x}\,dx = 2\ln|x|\)

\(\dfrac{x^3}{3}+2\ln|x|+C\)

Answer: \(\dfrac{x^3}{3}+2\ln|x|+C\)

5
\(f'(x)=e^x+2\) and \(f(0)=4\). Find \(f(x)\).
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\(f(x)=e^x+2x+C\)

\(f(0)=1+C\)

\(1+C=4\)

\(C=3\)

\(f(x)=e^x+2x+3\)

Answer: \(e^x+2x+3\)

Level 3

Hard

Rewrite first, or match a scaled inverse-trig form.

Example
Find \(\int \left(3x^2-\dfrac{6}{x^2}+2\right)\,dx\).

\(-\dfrac{6}{x^2} = -6x^{-2}\)

\(\int -6x^{-2}\,dx = -6\cdot \dfrac{x^{-1}}{-1}\)

\(\dfrac{6}{x}\)

\(x^3+\dfrac{6}{x}+2x+C\)

Answer: \(x^3+\dfrac{6}{x}+2x+C\)

1
Find \(\int \dfrac{1}{9+x^2}\,dx\).
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\(\int \dfrac{1}{a^2+x^2}\,dx = \dfrac{1}{a}\arctan\dfrac{x}{a}+C\)

\(a=3\)

\(\dfrac{1}{3}\arctan\dfrac{x}{3}+C\)

Answer: \(\dfrac{1}{3}\arctan\dfrac{x}{3}+C\)

2
Find \(\int \dfrac{1}{\sqrt{4-x^2}}\,dx\).
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The derivative of \(\arcsin(x/2)\) is \(\dfrac{1}{\sqrt{4-x^2}}\).

\(\arcsin\dfrac{x}{2}+C\)

Answer: \(\arcsin\dfrac{x}{2}+C\)

3
\(f''(x)=12x\), \(f'(0)=5\), and \(f(0)=-2\). Find \(f(x)\).
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\(f'(x)=6x^2+C\)

\(f'(0)=C=5\)

\(f'(x)=6x^2+5\)

\(f(x)=2x^3+5x+D\)

\(f(0)=D=-2\)

\(f(x)=2x^3+5x-2\)

Answer: \(2x^3+5x-2\)

4
Which expression is an antiderivative of \(\dfrac{2}{x}\) for \(x<0\)?
A. \(2\ln|x|\)
B. \(2\ln x\)
C. \(-\dfrac{2}{x^2}\)
D. \(\ln|2x|\)
Show Answer

The domain is negative, so a bare \(\ln x\) is not defined.

The absolute value keeps the logarithm defined.

The factor 2 stays in front. It is not the same as \(\ln|2x|=\ln 2+\ln|x|\).

Answer: A. \(2\ln|x|\)

5
Find \(\int (\csc^2 x - 3\sin x)\,dx\).
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\(\int \csc^2 x\,dx = -\cot x\)

\(\int \sin x\,dx = -\cos x\)

\(-\cot x - 3(-\cos x) + C\)

\(-\cot x+3\cos x+C\)

Answer: \(-\cot x+3\cos x+C\)

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