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IB Mathematics: Analysis and Approaches HL — Topic 5 Calculus

Anti-Differentiation and Boundary Conditions

Reverse differentiation carefully, keep the constant of integration, and use a boundary condition to select one particular function.

AA HL · SL 5.5

Learning goal

Find general anti-derivatives using the power rule and determine particular anti-derivatives from boundary conditions.

Syllabus link

IB Mathematics AA HL: SL 5.5. This lesson develops the anti-differentiation and boundary-condition strand. The AA learning path develops definite integrals and areas later, after the standard-integrals material.

Big idea

Differentiation loses constants. Anti-differentiation must therefore produce a family of functions until extra information determines the vertical shift.

Key relationship

\(\displaystyle \int f(x)\,dx=F(x)+C\) when \(F' = f\).

1

What is anti-differentiation?

Differentiation takes a function and finds its derivative. Anti-differentiation reverses that process: from a derivative, we recover a function whose derivative is the expression we started with.

Definition

If \(F'(x)=f(x)\), then \(F(x)\) is an anti-derivative of \(f(x)\). We write

\[\int f(x)\,dx=F(x)+C.\]

The symbol \(\int\) is the integral sign and \(dx\) tells us the variable of integration.

Key vocabulary: integrand

The integrand is the expression being integrated: the part after the integral sign and before \(dx\).

\[\int \underbrace{f(x)}_{\text{integrand}}\,dx=F(x)+C.\]

For example, in \(\int(4x^3-6x+7)\,dx\), the integrand is \(4x^3-6x+7\).

Quick example

Since \(\dfrac{d}{dx}(x^3)=3x^2\), one anti-derivative of \(3x^2\) is \(x^3\). Therefore

\[\int 3x^2\,dx=x^3+C.\]
2

Why do we need \(+C\)?

The derivative of a constant is zero. Differentiation therefore cannot distinguish between functions that differ only by a vertical shift.

Graphs of y equals x cubed and vertical shifts x cubed plus or minus 3
The curves \(y=x^3\), \(y=x^3+3\) and \(y=x^3-3\) all have derivative \(3x^2\).

Important habit

For an indefinite integral, include \(+C\) unless a condition is supplied that allows you to determine \(C\).

3

The basic power rule

Power rule for anti-differentiation

\[\boxed{\int x^n\,dx=\frac{x^{n+1}}{n+1}+C},\qquad n\ne -1.\]

Increase the power by 1, then divide by the new power.

IntegrandAnti-derivativeReason
\(x^2\)\(\dfrac{x^3}{3}+C\)Add 1 to the power, then divide by 3.
\(x\)\(\dfrac{x^2}{2}+C\)Think of \(x\) as \(x^1\).
\(1\)\(x+C\)Think of \(1\) as \(x^0\).
\(x^{-2}\)\(-x^{-1}+C\)Add 1: \(-2\to-1\), then divide by \(-1\).
\(5x^3\)\(\dfrac{5x^4}{4}+C\)Keep the coefficient and apply the power rule.

Worked example 1: one term

Find \(\int 6x^2\,dx\).

Solution

\[\int 6x^2\,dx=6\left(\frac{x^3}{3}\right)+C=2x^3+C.\]

Check: \(\dfrac{d}{dx}(2x^3+C)=6x^2\).

The exception

The rule above does not work for \(n=-1\), because it would require division by zero. The integral of \(1/x\) is treated separately later in Topic 5.

4

Worked examples

Worked example 2: several polynomial terms

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

Solution

\[\begin{aligned}\int(4x^3-6x+7)\,dx&=\int4x^3\,dx-\int6x\,dx+\int7\,dx\\&=x^4-3x^2+7x+C.\end{aligned}\]

Worked example 3: negative powers

Find \(\int\left(3x^2-\dfrac4{x^2}+5\right)\,dx\).

Solution

Rewrite \(-4/x^2\) as \(-4x^{-2}\):

\[\begin{aligned}\int\left(3x^2-\frac4{x^2}+5\right)dx&=\int(3x^2-4x^{-2}+5)dx\\&=x^3+4x^{-1}+5x+C\\&=x^3+\frac4x+5x+C.\end{aligned}\]

Worked example 4: from derivative notation

Given \(F'(x)=12x^3-8x+1\), find the general form of \(F(x)\).

Solution

Because \(F'(x)\) is the derivative, integrate to recover \(F(x)\):

\[F(x)=3x^4-4x^2+x+C.\]

Worked example 5: a context without a boundary condition

A particle has velocity \(v(t)=3t^2-2t\). Find a general expression for its position \(s(t)\).

Solution

Velocity is the derivative of position, so \(s'(t)=v(t)\). Therefore

\[s(t)=\int(3t^2-2t)dt=t^3-t^2+C.\]

The unknown \(C\) represents the starting position.

5

How to check an anti-derivative

Differentiate your answer

If you claim that \(F(x)\) is an anti-derivative of \(f(x)\), differentiate \(F\). You should recover the original integrand exactly.

\[F(x)=3x^4-4x^2+x+C\quad\Rightarrow\quad F'(x)=12x^3-8x+1.\]
6

Common traps

Things to check every time

  • Forgetting \(+C\): an indefinite integral represents a family of functions.
  • Not rewriting powers: expressions such as \(1/x^2\) are usually easier to integrate as \(x^{-2}\).
  • Using the differentiation power rule backwards incorrectly: for integration, add 1 to the exponent before dividing.
  • Trying to use the power rule on \(x^{-1}\): that case needs the logarithm rule.
7

Quick recap: indefinite integrals give a family of functions

An indefinite integral does not usually produce one unique function. It produces every vertical translation whose derivative is the required function.

\[\int f(x)\,dx=F(x)+C.\]

To identify one particular member of the family, we need additional information.

8

What is a boundary condition?

Boundary condition

A boundary condition gives the value of the function at a particular input, for example \(y=5\) when \(x=1\), or \(F(0)=3\). It allows the constant \(C\) to be determined.

9

General method

Five-step method

  1. Integrate to find the general anti-derivative.
  2. Include \(+C\).
  3. Substitute the given boundary condition into the anti-derivative.
  4. Solve for \(C\).
  5. Write the particular anti-derivative and check it by differentiating and substituting the condition.

Key habit

Do not substitute the condition into the derivative. First integrate, then substitute the condition into the anti-derivative.

10

Visual idea: one condition selects one curve

The derivative determines the shape of a family. The constant \(C\) shifts the graph vertically. A boundary condition selects the one curve through the stated point.

Family of cubic anti-derivatives with one curve passing through the boundary condition 1 comma 5
Only \(y=2x^3+3\) passes through \((1,5)\), so the condition forces \(C=3\).
11

Worked examples with boundary conditions

Worked example 1: \(dy/dx\) notation

Given \(\dfrac{dy}{dx}=6x^2\) and \(y=5\) when \(x=1\), find \(y\).

Solution

\[y=\int6x^2dx=2x^3+C.\]

Use the condition:

\[5=2(1)^3+C\Rightarrow C=3.\]

Answer \(y=2x^3+3\).

Worked example 2: function notation

Given \(F'(x)=4x^3-6x+2\) and \(F(0)=3\), find \(F(x)\).

Solution

\[F(x)=x^4-3x^2+2x+C,\qquad 3=C.\]

Answer \(F(x)=x^4-3x^2+2x+3\).

Worked example 3: curve through a point

Given \(\dfrac{dy}{dx}=3x^2-4x+1\) and the curve passes through \((2,7)\), find the equation of the curve.

Solution

\[y=x^3-2x^2+x+C.\]

Substitute \((2,7)\): \(7=8-8+2+C\), so \(C=5\).

Answer \(y=x^3-2x^2+x+5\).

Worked example 4: velocity to position

A particle has velocity \(v(t)=6t-4\) and position \(s(0)=10\). Find \(s(t)\).

Solution

\[s(t)=3t^2-4t+C,\qquad C=10.\]

Answer \(s(t)=3t^2-4t+10\).

Worked example 5: negative powers

Given \(F'(x)=2x^{-3}+5\) and \(F(1)=4\), find \(F(x)\).

Solution

\[F(x)=-x^{-2}+5x+C=5x-\frac1{x^2}+C.\]

At \(x=1\), \(4=5-1+C\), so \(C=0\).

Answer \(F(x)=5x-\dfrac1{x^2}\).

12

Common traps with boundary conditions

Boundary-condition traps

  • Do not forget \(+C\) before applying the condition.
  • Substitute the condition into the anti-derivative, not the derivative.
  • If the condition is given as a point \((a,b)\), use \(x=a\) and \(y=b\).
  • After finding \(C\), write the final particular function explicitly.
  • Check both the derivative and the condition.
13

Practice

Part A — General anti-derivatives and \(+C\)

Basic powers

  1. \(\int x^4dx\)
  2. \(\int7x^6dx\)
  3. \(\int5dx\)
  4. \(\int-3x^2dx\)
  5. \(\int x^{-3}dx\)
  6. \(\int9x^{-4}dx\)

Sums and differences

  1. \(\int(6x^2+4x-1)dx\)
  2. \(\int(10x^4-3x^2+8)dx\)
  3. \(\int(2x^3-6/x^2)dx\)
  4. \(\int(5x^2+4/x^3-7)dx\)
  5. Find \(F(x)\) if \(F'(x)=15x^2-4x+6\).
  6. Find \(s(t)\) if \(v(t)=6t^2+2t-5\).

Part B — Boundary conditions

  1. \(dy/dx=4x\), and \(y=9\) when \(x=2\).
  2. \(F'(x)=9x^2\), \(F(1)=10\).
  3. \(dy/dx=5x^4-2\), and \(y=0\) when \(x=1\).
  4. \(F'(x)=8x^3-6x\), \(F(0)=-4\).
  5. \(dy/dx=3x^2+4x-1\), curve through \((1,6)\).
  6. \(F'(x)=-6x^{-4}+2x\), \(F(1)=5\).
  7. \(F'(x)=12x^3-4x+1\), \(F(0)=-2\).
  8. A curve has gradient \(2x-3\) and passes through \((4,1)\).
  9. \(v(t)=4t+3\), \(s(0)=2\).
  10. \(a(t)=6t\), \(v(0)=-1\).
  11. \(F'(x)=5x^2-4x^{-3}\), \(F(2)=3\).
  12. A student writes \(y=2x^3+4\) for \(dy/dx=6x^2\), \(y=5\) at \(x=1\). Explain the error.
14

Answer key

Part A

1. \(x^5/5+C\)

2. \(x^7+C\)

3. \(5x+C\)

4. \(-x^3+C\)

5. \(-1/(2x^2)+C\)

6. \(-3/x^3+C\)

7. \(2x^3+2x^2-x+C\)

8. \(2x^5-x^3+8x+C\)

9. \(x^4/2+6/x+C\)

10. \(5x^3/3-2/x^2-7x+C\)

11. \(F=5x^3-2x^2+6x+C\)

12. \(s=2t^3+t^2-5t+C\)

Part B

1. \(y=2x^2+1\)

2. \(F=3x^3+7\)

3. \(y=x^5-2x+1\)

4. \(F=2x^4-3x^2-4\)

5. \(y=x^3+2x^2-x+4\)

6. \(F=2/x^3+x^2+2\)

7. \(F=3x^4-2x^2+x-2\)

8. \(y=x^2-3x-3\)

9. \(s=2t^2+3t+2\)

10. \(v=3t^2-1\)

11. \(F=\frac53x^3+\frac2{x^2}-\frac{65}{6}\)

12. The proposed function gives 6, not 5, at \(x=1\); correct: \(y=2x^3+3\).

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