Anti-differentiate by recognition
When the derivative of a function is supplied, use that relation to identify an anti-derivative directly. If the target is a constant multiple of the supplied derivative, keep the same multiple on the function
\[F'(x)=f(x)\Rightarrow\int f(x)\,dx=F(x)+C\]
- Read the supplied relation A supplied derivative gives an anti-derivative without integrating its visible factors separately \(\displaystyle F'(x)=f(x)\Rightarrow\int f(x)\,dx=F(x)+C\)
- Carry a constant scale A constant multiplying the derivative also multiplies the recognised function \(\displaystyle g(x)=2F'(x)\Rightarrow\int g(x)\,dx=2F(x)+C\)
- Use one additive constant The complete family has one arbitrary constant for the whole anti-derivative \(\displaystyle \int\left(f(x)+h(x)\right)\,dx=F(x)+H(x)+C\)
- Differentiate to check Differentiate the proposed primitive and compare it with the target function \(\displaystyle \frac{d}{dx}\left(2F(x)+C\right)=2F'(x)\)
Worked example
Given the function and relation below, find the complete family of anti-derivatives of g
\[F(x)=2\left(x^2+3\right)^{3/2},\quad g(x)=3F'(x)\]
- \(\displaystyle F'(x)=\frac{d}{dx}\left(2\left(x^2+3\right)^{3/2}\right)=6x\sqrt{x^2+3}\)Differentiate the supplied function to make the derivative relation explicit
- \(\displaystyle g(x)=3F'(x)\)The target is three times the supplied derivative
- \(\displaystyle \int g(x)\,dx=3\int F'(x)\,dx=3F(x)+C\)Recognition returns F(x), while the scale factor 3 is preserved
- \(\displaystyle \int g(x)\,dx=6\left(x^2+3\right)^{3/2}+C\)Substitute the supplied expression for F(x) into the recognised family
Common mistakes
Using the derivative as the anti-derivative
✕\(\displaystyle \int g(x)\,dx=3F'(x)+C\)✓\(\displaystyle \int g(x)\,dx=3F(x)+C\)
The supplied derivative identifies the function F, so integration returns F rather than F'Reversing the scale factor
✕\(\displaystyle g(x)=4F'(x)\Rightarrow\int g(x)\,dx=\frac{1}{4}F(x)+C\)✓\(\displaystyle g(x)=4F'(x)\Rightarrow\int g(x)\,dx=4F(x)+C\)
A constant multiplying the derivative remains the same constant multiplying the recognised functionIntegrating visible factors separately
✕\(\displaystyle \int 6x\sqrt{x^2+3}\,dx=3x^2+\frac{2}{3}\left(x^2+3\right)^{3/2}+C\)✓\(\displaystyle \int 6x\sqrt{x^2+3}\,dx=2\left(x^2+3\right)^{3/2}+C\)
A supplied derivative is recognised as one derivative instead of being anti-differentiated factor by factorDifficulty levels
introRecognise a polynomial-power derivative and write its complete anti-derivative family
standardUse a quadratic inner polynomial and a non-unit scale factor
challengeRecognise an exponential-polynomial derivative or determine a specific family member
examSelect the recognised primitive and complete an independent coefficient check
Questions about this topic
How is recognition different from applying the power rule?+
Recognition uses a supplied derivative relation to identify the original function. The visible expression is not split into separate factors for integration
Why is the same scale factor used after integration?+
Differentiation and anti-differentiation are linear, so a constant multiplying a derivative also multiplies its recognised function
How can a recognised anti-derivative be checked?+
Differentiate the proposed function and compare the result with the target integrand, including its scale factor
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