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Anti-differentiate exponential and circular functions

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Reuse the primitive rules inside parts

The elementary primitive rules supply v when dv contains an exponential, sine or cosine with a linear argument. Check the reciprocal scale and circular sign before applying the parts formula. In coefficient tables, keep the original term order and signed arguments even when a parity identity gives an equivalent expression

Year 12, VCE Units 3 & 4, Chapter 5, 4 difficulty levels, No calculatorBuild a worksheet

Anti-differentiate exponential and circular functions

For an exponential or circular function with a linear argument, divide by the coefficient of the variable. Sine changes to negative cosine, cosine changes to sine, and the exponent of an exponential stays unchanged. Here k≠0, and each complete family includes one arbitrary constant

\[\int e^{kx}\,dx=\frac{1}{k}e^{kx}+C,\quad\int\sin(kx)\,dx=-\frac{1}{k}\cos(kx)+C,\quad\int\cos(kx)\,dx=\frac{1}{k}\sin(kx)+C\]
  1. Keep the exponential argument Divide the coefficient by k and leave the exponent kx unchanged \(\displaystyle \int4e^{2x}\,dx=2e^{2x}+C\)
  2. Keep the sign of a negative rate A negative coefficient of x produces a negative reciprocal factor \(\displaystyle \int6e^{-3x}\,dx=-2e^{-3x}+C\)
  3. Change sine to negative cosine The sine anti-derivative includes both the reciprocal factor and a negative sign \(\displaystyle \int4\sin(2x)\,dx=-2\cos(2x)+C\)
  4. Change cosine to sine The cosine anti-derivative changes the function and includes the reciprocal factor \(\displaystyle \int3\cos(3x)\,dx=\sin(3x)+C\)

Worked example

Find the complete family of anti-derivatives of the function below

\[\int6\sin(2x)\,dx\]
  1. \(\displaystyle a=6,\quad k=2\)The coefficient is 6 and the sine argument has inner coefficient 2.
  2. \(\displaystyle \frac{a}{k}=\frac{6}{2}=3\)Divide the coefficient by the inner coefficient
  3. \(\displaystyle \int6\sin(2x)\,dx=-3\cos(2x)+C\)Sine changes to negative cosine, so the coefficient is −3.
  4. \(\displaystyle \frac{d}{dx}\left(-3\cos(2x)+C\right)=6\sin(2x)\)Differentiation returns the original integrand, which checks the result

Common mistakes

Omitting the reciprocal factor
✕\(\displaystyle \int6\sin(2x)\,dx=-6\cos(2x)+C\)✓\(\displaystyle \int6\sin(2x)\,dx=-3\cos(2x)+C\)
The coefficient must be divided by the inner coefficient 2 before the sine sign is applied
Using the wrong circular sign
✕\(\displaystyle \int3\sin(2x)\,dx=\frac{3}{2}\cos(2x)+C\)✓\(\displaystyle \int3\sin(2x)\,dx=-\frac{3}{2}\cos(2x)+C\)
The derivative of cosine is negative sine, so the anti-derivative of sine needs a negative sign
Changing the exponential argument
✕\(\displaystyle \int4e^{2x}\,dx=2e^{4x}+C\)✓\(\displaystyle \int4e^{2x}\,dx=2e^{2x}+C\)
Anti-differentiation changes the coefficient but keeps the exponential argument 2x

Difficulty levels

introAnti-differentiate one exponential or circular term with a whole reciprocal factor
standardHandle a signed linear combination, a rule choice or a specific-member condition
challengeAnti-differentiate a mixed exponential and circular integrand or evaluate a conditioned member exactly
examChoose the correct primitive from distractors involving reciprocal factors, signs and function changes

Questions about this topic

Why is the coefficient divided by k?+

Differentiating an exponential with exponent kx, sine(kx) or cosine(kx) creates a factor k. Dividing by k removes that extra factor

Why does the anti-derivative of sine have a negative sign?+

The derivative of cos(kx) is −k sin(kx), so −cos(kx) divided by k differentiates to sin(kx)

What happens to an exponential's exponent when anti-differentiating?+

The exponent remains kx. Only the coefficient is changed by division by k.

Can I use a calculator?+

No. This topic is practised without a calculator

Which year level and chapter is this?+

Year 12, VCE Units 3 & 4, Specialist Mathematics, chapter 5: Integral calculus

In this chapter

Integrate inverse circular kernels and change variablesAnti-differentiate reciprocal and linear-reciprocal functionsIntegration by recognition with logarithmic componentsAnti-differentiate by recognition from a supplied derivativeIntegrate using partial fractions, identities and partsDecompose rational expressions into partial fractionsCompound, double-angle and reciprocal identitiesAnti-differentiate exponential and circular functionsFind exact areas and assess numerical integration resultsFind the area between two curvesApproximate area by an endpoint or trapezium ruleFind areas involving reciprocal curvesAudit numerical integration loops and error enclosuresSet up volumes, parametric lengths and surfaces of revolution