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IB Mathematics: Analysis and Approaches SL/HL — Topic 5 Calculus

Standard Integrals

Recognise standard antiderivatives, use linearity, rewrite powers cleanly and handle linear inputs with the correct scale factor.

AA SL/HL · SL 5.10

Learning goal

Recognise and use the standard integrals needed in AA SL, including powers, \(1/x\), \(e^x\), sine and cosine.

Syllabus link

SL 5.10 Standard indefinite integrals and simple results involving linear inputs.

Big idea

Integration reverses differentiation. Check answers by differentiating them.

Boundary

This lesson covers standard forms and linear inputs. Non-linear reverse-chain-rule substitution is treated separately.

1

Core standard integrals

The following standard results should become automatic. For indefinite integrals, always include the constant of integration \(C\).

Flow diagram showing differentiation and integration as reverse processes
Integration reverses differentiation. The antiderivative includes an unknown constant.
IntegralResult and notes
\(\int k\,dx\)\(kx+C\), where \(k\) is constant.
\(\int x^n\,dx\)\(\dfrac{x^{n+1}}{n+1}+C\), for \(n\in\mathbb Q\), \(n\ne-1\).
\(\int\frac1x\,dx\)\(\ln|x|+C\). If \(x>0\), this is \(\ln x+C\).
\(\int e^x\,dx\)\(e^x+C\).
\(\int\sin x\,dx\)\(-\cos x+C\).
\(\int\cos x\,dx\)\(\sin x+C\).

Important exception

The power rule does not work for \(n=-1\), because \(n+1=0\) would require division by zero. Use the separate logarithmic result:

\[\int x^{-1}\,dx=\int\frac1x\,dx=\ln|x|+C.\]

Radians reminder

The derivative and antiderivative results for \(\sin x\) and \(\cos x\) are stated in radians. In calculus, trigonometric functions are interpreted in radians unless explicitly stated otherwise.

2

Linearity: combining standard integrals

Linearity lets us integrate term by term. If \(a\) and \(b\) are constants, then

\[\int\big(af(x)+bg(x)\big)\,dx=a\int f(x)\,dx+b\int g(x)\,dx.\]

Split sums and differences, and keep constant multipliers in front.

Flow diagram combining two standard antiderivatives into the antiderivative of a linear combination

Useful simplification step

Before integrating, rewrite roots and fractions as powers when this makes the standard table easier to use:

\[\sqrt{x}=x^{1/2},\qquad\frac1{\sqrt{x}}=x^{-1/2},\qquad\frac1{x^3}=x^{-3}.\]

Worked example 1: powers, logarithms and exponentials

Find \(\int\left(6x^2-\frac4x+5e^x\right)\,dx\).

Solution

Use linearity to separate the three standard integrals:

\[\begin{aligned}\int\left(6x^2-\frac4x+5e^x\right)\,dx&=6\int x^2\,dx-4\int\frac1x\,dx+5\int e^x\,dx\\&=6\cdot\frac{x^3}{3}-4\ln|x|+5e^x+C\\&=\boxed{2x^3-4\ln|x|+5e^x+C}.\end{aligned}\]

Worked example 2: trigonometric standard integrals

Find \(\int(3\sin x-2\cos x)\,dx\).

Solution

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

\[\begin{aligned}\int(3\sin x-2\cos x)\,dx&=3\int\sin x\,dx-2\int\cos x\,dx\\&=\boxed{-3\cos x-2\sin x+C}.\end{aligned}\]

Worked example 3: rewriting roots as powers

Find \(\int\left(4x^3-3\sqrt{x}+\frac2{\sqrt{x}}\right)\,dx\).

Solution

First write \(\sqrt{x}=x^{1/2}\) and \(1/\sqrt{x}=x^{-1/2}\). Apply the power rule to each term:

\[\begin{aligned}\int\left(4x^3-3x^{1/2}+2x^{-1/2}\right)\,dx&=x^4-3\frac{x^{3/2}}{3/2}+2\frac{x^{1/2}}{1/2}+C\\&=\boxed{x^4-2x^{3/2}+4x^{1/2}+C}.\end{aligned}\]
3

Linear inputs: functions of \(ax+b\)

Once the standard results and linearity are secure, we can change the input from \(x\) to a linear expression \(ax+b\).

Linear-input rule

If \(F'(x)=f(x)\) and \(a\ne0\), then

\[\boxed{\int f(ax+b)\,dx=\frac1aF(ax+b)+C}.\]

Integrate as usual, then divide by the coefficient of x inside the bracket.

Flow diagram from a standard integral to a linear-input integral, dividing by the inner coefficient a

Why divide by a?

Check by differentiating:

\[\frac{d}{dx}\left(\frac1aF(ax+b)\right)=\frac1a\,F'(ax+b)\,a=f(ax+b).\]

The factor \(1/a\) cancels the extra \(a\) produced by the chain rule.

IntegralResult (with a ≠ 0)
\(\int(ax+b)^n\,dx\)\(\dfrac{(ax+b)^{n+1}}{a(n+1)}+C\), \(n\ne-1\).
\(\int\frac1{ax+b}\,dx\)\(\dfrac1a\ln|ax+b|+C\).
\(\int e^{ax+b}\,dx\)\(\dfrac1ae^{ax+b}+C\).
\(\int\sin(ax+b)\,dx\)\(-\dfrac1a\cos(ax+b)+C\).
\(\int\cos(ax+b)\,dx\)\(\dfrac1a\sin(ax+b)+C\).

Worked example 4: cosine with a linear input

Find \(\int\cos(2x+3)\,dx\).

Solution

Since \(\int\cos u\,du=\sin u+C\), and the coefficient of \(x\) inside is \(2\),

\[\int\cos(2x+3)\,dx=\boxed{\frac12\sin(2x+3)+C}.\]

Worked example 5: logarithmic integral with a negative coefficient

Find \(\int\frac1{5-2x}\,dx\).

Solution

Here \(5-2x=-2x+5\), so \(a=-2\). Use the logarithmic linear-input rule:

\[\int\frac1{5-2x}\,dx=\boxed{-\frac12\ln|5-2x|+C}.\]

Check.

\[\frac{d}{dx}\left(-\frac12\ln|5-2x|\right)=-\frac12\cdot\frac{-2}{5-2x}=\frac1{5-2x}.\]

Worked example 6: mixed standard and linear inputs

Find \(\int\left(4x^3-3\sqrt{x}+2e^{1-3x}\right)\,dx\).

Solution

First write \(\sqrt{x}=x^{1/2}\). Integrate term by term; the exponential input has coefficient \(-3\).

\[\begin{aligned}\int\left(4x^3-3x^{1/2}+2e^{1-3x}\right)\,dx&=x^4-3\frac{x^{3/2}}{3/2}+2\left(-\frac13e^{1-3x}\right)+C\\&=\boxed{x^4-2x^{3/2}-\frac23e^{1-3x}+C}.\end{aligned}\]

Worked example 7: power of a linear expression

Find \(\int(3x-5)^7\,dx\).

Solution

The inside is \(3x-5\), so \(a=3\). Increase the power to \(8\), then divide by both \(8\) and \(3\).

\[\int(3x-5)^7\,dx=\frac{(3x-5)^8}{3\cdot8}+C=\boxed{\frac{(3x-5)^8}{24}+C}.\]

Check.

\[\frac{d}{dx}\left(\frac{(3x-5)^8}{24}\right)=\frac1{24}\cdot8(3x-5)^7\cdot3=(3x-5)^7.\]
4

Common traps

TrapFix
Forgetting \(+C\)Include it for every indefinite integral unless a condition determines a particular antiderivative.
Using power rule for \(x^{-1}\)Use \(\ln|x|+C\).
Forgetting linear coefficientFor \(f(ax+b)\), divide by \(a\).
Dividing by the constant termDivide by the coefficient of \(x\), not \(b\).
Degrees instead of radiansCalculus uses radians unless stated otherwise.
Dropping absolute valuesWrite \(\ln|ax+b|\) unless the domain guarantees positivity.

Check habit

Differentiate your answer. It should reconstruct the original integrand.

5

Practice questions

Try these without looking at the answers first. For indefinite integrals, include \(+C\).

A. Core standard integrals and linearity

  1. \(\int7x^6dx\)
  2. \(\int(5x^4-3x^2+8)dx\)
  3. \(\int(6/x+4e^x)dx\)
  4. \(\int(3\sin x-2\cos x)dx\)
  5. \(\int(\sqrt{x}+2/\sqrt{x})dx\)

B. Linear inputs

  1. \(\int(2x+1)^5dx\)
  2. \(\int(7-3x)^4dx\)
  3. \(\int e^{4x-1}dx\)
  4. \(\int\sin(5x)dx\)
  5. \(\int\cos(3x-\pi)dx\)
  6. \(\int\frac1{4x+9}dx\)
  7. \(\int\frac3{2-5x}dx\)

C. Mixed and particular antiderivatives

  1. \(\int(2(3x-1)^4-5e^{-x})dx\)
  2. \(\int(6x^2+4\cos(2x)-3/x)dx\)
  3. \(\int(5e^{2x-1}-2\sin(3x))dx\)
  4. Find \(F\) if \(F'(x)=6x^2-2/x\) and \(F(1)=4\).
  5. Explain why \(\int x^{-1}dx\) is not found using \(x^0/0\).
6

Answer key

Show the answers to A, B and C

A

  1. \(x^7+C\)
  2. \(x^5-x^3+8x+C\)
  3. \(6\ln|x|+4e^x+C\)
  4. \(-3\cos x-2\sin x+C\)
  5. \(\frac23x^{3/2}+4x^{1/2}+C\)

B

  1. \((2x+1)^6/12+C\)
  2. \(-(7-3x)^5/15+C\)
  3. \(e^{4x-1}/4+C\)
  4. \(-\cos(5x)/5+C\)
  5. \(\sin(3x-\pi)/3+C\)
  6. \(\ln|4x+9|/4+C\)
  7. \(-3\ln|2-5x|/5+C\)

C

  1. \(2(3x-1)^5/15+5e^{-x}+C\)
  2. \(2x^3+2\sin(2x)-3\ln|x|+C\)
  3. \(\frac52e^{2x-1}+\frac23\cos(3x)+C\)
  4. First integrate: \(F(x)=2x^3-2\ln|x|+C\). Since \(F(1)=4\), we have \(2-2\ln1+C=4\), so \(C=2\). Therefore \(F(x)=2x^3-2\ln|x|+2\).
  5. The power rule is invalid because \(n+1=0\); use \(\ln|x|+C\).

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