Integration Rules and Composite Functions

Extending the Basic Rules to Composite Functions

The standard integration rules apply when the argument is simply x. But what happens when we need to integrate something like (2x + 5)⁶, or e^3x−1? In differentiation, the chain rule handles these cases. In integration, we apply the chain rule in reverse, and the result is a family of rules for linear composite functions.

A linear composite function has an inner function of the form (ax + b), where a and b are constants. The key insight is that when we differentiate F(ax + b), the chain rule produces a factor of a (the derivative of the inner function). So when integrating, we must compensate by dividing by a.

The Power Rule for (ax + b)

For ∫ (ax + b)ⁿ dx, raise the power by 1 (giving n + 1), divide by the new power (n + 1) and divide by a, the coefficient of x:

∫ (ax+b)ⁿ dx = (ax+b)ⁿ⁺¹ / [a(n+1)] + c

For example, ∫ (4x − 1)³ dx = (4x−1)⁴/[4 × 4] + c = (4x−1)⁴/16 + c. Verify: d/dx[(4x−1)⁴/16] = (1/16) × 4(4x−1)³ × 4 = (4x−1)³ ✓

This rule does not apply when n = −1 (the special case handled by the logarithm rule below).

Exponential Functions: e^ax+b

Since d/dx[e^ax+b] = a e^ax+b, integrating reverses this and we divide by a:

∫ e^ax+b dx = (1/a) e^ax+b + c

For example, ∫ e^5x+2 dx = (1/5)e^5x+2 + c. Notice that when a = 1, this reduces to ∫ e^x dx = e^x + c, which is consistent with the basic rule.

The Logarithm Rule for 1/(ax + b)

This covers the n = −1 case for linear functions:

∫ 1/(ax+b) dx = (1/a) ln|ax+b| + c

The absolute value is required because ln is only defined for positive arguments, yet 1/(ax+b) is defined for all x where ax + b ≠ 0. For example, ∫ 1/(3x−2) dx = (1/3) ln|3x−2| + c.

Trigonometric Rules for (ax + b)

Reversing the chain rule for trig functions:

∫ sin(ax+b) dx = −(1/a) cos(ax+b) + c     ∫ cos(ax+b) dx = (1/a) sin(ax+b) + c

The negative sign in the sine integral often causes errors. Remember: d/dx[−cos(ax+b)] = a sin(ax+b), so ∫ sin(ax+b) dx = −cos(ax+b)/a + c. For cosine: d/dx[sin(ax+b)] = a cos(ax+b), so ∫ cos(ax+b) dx = sin(ax+b)/a + c.

Putting It Together: Multi-Term Integrals

Integrals often combine several of these rules. Apply each rule to the corresponding term, then add the single constant c at the end (not one c per term).

Example: ∫ [(2x+1)³ + e^−x + cos(3x)] dx = (2x+1)⁴/8 − e^−x + (1/3)sin(3x) + c

Always perform a differentiation check on your final answer before moving on — this is the single most effective error-catching strategy in integration.