Additional Mathematics Synthesis Guide 30: Integration Rule Selection — Rational Powers, Exponentials, Trigonometric Forms and Reverse Chain Rule

Rewrite → recognise derivative pattern → compensate for inner factor → integrate → verify by differentiation → apply limits exactly.

1. The power antiderivative

For n≠−1,

∫xⁿ dx = xⁿ⁺¹/(n+1)+C.

The rule is checked by differentiating the result: the new exponent n+1 comes down and cancels the denominator.

2. Rational powers are ordinary powers

For

∫√x dx,

write √x=x^1/2:

∫x^1/2dx=(2/3)x^3/2+C.

Radical notation can hide a power-rule problem.

3. Negative powers

For

∫4/x³ dx,

write 4x⁻³:

∫4x⁻³dx=−2x⁻²+C=−2/x²+C.

Keep the original domain x≠0.

4. The exceptional power n=−1

The power formula would divide by zero when n=−1. Instead,

∫1/x dx=ln|x|+C

on intervals not crossing x=0.

If this logarithmic antiderivative is outside the immediate assessed form being practised, treat it as a structural exception rather than forcing the power rule.

5. Exponential antiderivatives

Since d(eˣ)/dx=eˣ,

∫eˣ dx=eˣ+C.

For a linear inner function:

∫e^3x−2dx=(1/3)e^3x−2+C.

The factor 1/3 compensates for the derivative of 3x−2.

6. Reverse chain rule

If differentiation of F(ax+b) produces an extra factor a, then integration must divide by a.

For example:

∫(2x+1)⁵ dx=(2x+1)⁶/12+C.

Differentiate to check:

(1/12)·6(2x+1)⁵·2=(2x+1)⁵.

7. General linear-power form

For n≠−1 and a≠0,

∫(ax+b)ⁿ dx=(ax+b)ⁿ⁺¹/[a(n+1)]+C.

This is the power rule plus reverse chain rule in one formula.

8. Sine and cosine antiderivatives

• ∫cosx dx=sinx+C
• ∫sinx dx=−cosx+C

The negative sign in the sine antiderivative is essential because d(−cosx)/dx=sinx.

9. Linear-angle trigonometric forms

Examples:

• ∫cos(4x+1)dx=(1/4)sin(4x+1)+C
• ∫sin(5x−2)dx=−(1/5)cos(5x−2)+C

These standard trigonometric calculus formulas use radian measure.

10. Secant-squared form

Since d(tanx)/dx=sec²x,

∫sec²x dx=tanx+C

on intervals where tanx is defined.

Thus

∫sec²(3x)dx=(1/3)tan(3x)+C.

11. Simplify before integrating

Evaluate

∫(x²+2x)/x dx, x≠0.

Simplify:

∫(x+2)dx=x²/2+2x+C

on the original domain x≠0.

12. Expand when it lowers complexity

For

∫(x+1)(x−2)dx,

expand to x²−x−2, then integrate term by term:

x³/3−x²/2−2x+C.

There is no product rule for integration corresponding directly to the differentiation product rule at this level, so algebraic simplification is often the right entry.

13. Definite integrals remove the arbitrary constant

Evaluate

∫₁³(2x+1)dx.

An antiderivative is x²+x. Thus

(9+3)−(1+1)=10.

The +C cancels between the two endpoint evaluations, so it is omitted in definite-integral working.

14. Keep exact trigonometric values

Evaluate

∫₀^π/3cosx dx.

The exact result is

sin(π/3)−sin0=√3/2.

Do not replace √3/2 with a decimal unless requested.

15. Signed accumulation versus area

A definite integral adds signed contributions. If the graph lies below the x-axis, the contribution is negative.

Therefore

∫₀^2πsinx dx=0

even though the total geometric area is not zero. Guide 11 develops this distinction more fully.

16. Reverse differentiation is the best verification

If you claim

∫sin(3x)dx=−(1/3)cos(3x)+C,

differentiate:

−(1/3)[−3sin(3x)]=sin(3x).

If the derivative does not reproduce the original integrand exactly, the antiderivative is wrong.

17. Integrating a sum is termwise

For

∫[3x²+2eˣ−4cosx]dx,

integrate each term:

x³+2eˣ−4sinx+C.

A single constant of integration is enough for the whole indefinite integral.

18. An integrand may hide a derivative pattern

For

∫6(3x+1)⁴dx,

we can recognise that d(3x+1)⁵/dx=15(3x+1)⁴. Therefore

∫6(3x+1)⁴dx=(2/5)(3x+1)⁵+C.

Checking by differentiation is faster than trying to memorise a separate rule for every coefficient pattern.

19. Constants may be reconstructed from conditions

If f′(x)=2x+3 and f(1)=5, then

f(x)=x²+3x+C.

Use f(1)=5:

1+3+C=5, so C=1.

Hence f(x)=x²+3x+1.

20. Integration and kinematics

If acceleration a(t) is known, integrating gives velocity plus a constant determined by an initial velocity. Integrating velocity gives displacement or position plus another constant.

Each constant carries physical state information that differentiation had removed.

This is why an indefinite integral is a family of functions, not one function.

21. Common failure patterns

• Using the power rule on x⁻¹ and dividing by zero.
• Forgetting the reciprocal inner coefficient in ∫(ax+b)ⁿdx.
• Integrating sinx to +cosx.
• Using degree mode while interpreting standard trigonometric calculus formulas.
• Forgetting +C in an indefinite integral.
• Adding a separate arbitrary constant to every term.
• Expanding an expression when a simpler reverse-chain pattern is already visible.
• Failing to verify an uncertain antiderivative by differentiation.

22. A reliable integration routine

1. Rewrite radicals and reciprocals as powers where useful.
2. Simplify or expand algebra only if it reduces the integration complexity.
3. Identify the derivative pattern: power, exponential, sine, cosine or sec².
4. Check whether a linear inner function requires compensation.
5. Integrate exactly and include +C for indefinite integrals.
6. Differentiate the result if there is any uncertainty.
7. For definite integrals, apply limits only after finding a correct antiderivative.
8. Return to sign, units and geometric/physical interpretation where relevant.

23. Practice set

1. Integrate x⁴.
2. Integrate √x.
3. Integrate 6/x³.
4. Integrate (2x+1)⁵.
5. Integrate eˣ.
6. Integrate e^4x.
7. Integrate cosx.
8. Integrate sinx.
9. Integrate cos(5x).
10. Integrate sin(2x−1).
11. Integrate sec²(3x).
12. Integrate 3x²+2eˣ−4cosx.
13. Evaluate ∫₀²(3x²+1)dx.
14. Evaluate ∫₀^π/2cosx dx.
15. Find f if f′(x)=4x−1 and f(0)=3.
16. Why is x⁻¹ exceptional for the power rule?
17. Why does ∫cos(4x)dx contain a factor 1/4?
18. Why is differentiation a good integration check?
19. Why does a definite integral not need +C?
20. Give one example where simplifying algebra before integration is useful.

Answers

1. x⁵/5+C.
2. (2/3)x^3/2+C.
3. −3/x²+C.
4. (2x+1)⁶/12+C.
5. eˣ+C.
6. (1/4)e^4x+C.
7. sinx+C.
8. −cosx+C.
9. (1/5)sin5x+C.
10. −(1/2)cos(2x−1)+C.
11. (1/3)tan3x+C.
12. x³+2eˣ−4sinx+C.
13. [x³+x]₀²=10.
14. 1.
15. f=x²·2? Correctly: ∫(4x−1)dx=2x²−x+C; C=3, so f=2x²−x+3.
16. The usual power formula would require division by n+1=0.
17. Differentiating sin(4x) produces an extra factor 4, so the antiderivative must divide by 4.
18. The derivative of an antiderivative must reproduce the original integrand exactly.
19. The same arbitrary constant appears at both endpoints and cancels.
20. For example, (x²+2x)/x simplifies to x+2 on x≠0.

24. What mastery looks like

Mastery means the learner recognises antiderivative families from structure, compensates correctly for linear inner functions, knows when algebraic rewriting makes the pattern clearer, distinguishes indefinite from definite integration and routinely verifies uncertain work by differentiation.

The transfer test is to mix powers, exponentials and trigonometric forms without section labels. If the learner can select the correct reverse-derivative pattern and explain the compensation factor rather than guessing from appearance, integration has become controlled.

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Continue through Batch 08

• Guide 29: Surds, Conjugates, Rationalising Denominators, Surd Equations and Exact Verification
• Guide 31: Increasing and Decreasing Functions, Derivative Sign Charts, Stationary Inflexions and Second-Derivative Decisions
• Guide 32: Transformed Trigonometric Graphs — Amplitude, Period, Midline, Tangent Structure and Modelling
• Additional Mathematics Directory
• BTT Mathematics Hub