What is the integral of e^(3x)?

(1/3)e^(3x) + c. Divide by the 3.

What is the integral of sin x?

−cos x + c. Check: −cos x differentiates to sin x.

What is the integral of cos 2x?

(1/2) sin 2x + c. Divide by the 2.

What is the integral of 1/(2x + 5)?

(1/2) ln(2x + 5) + c. Divide by the 2.

What is the integral of (x + 4)³?

(x + 4)⁴/4 + c. The number multiplying x is 1.

What is the integral of sec²x?

tan x + c. tan x differentiates to sec²x.

Integrating (ax + b)ⁿ, exponentials and trigonometric functions

In plain words

The chain rule multiplies by the derivative of the inside. Integration runs the film backwards, so it divides by it. That one idea gives five standard integrals, as long as the inside is a simple ax + b.

7 things to know

(ax + b)ⁿ integrates to (ax + b)ⁿ⁺¹ ÷ (a(n + 1)), for n ≠ −1.
1/(ax + b) integrates to (1/a) ln(ax + b).
e^(ax + b) integrates to (1/a)e^(ax + b).
sin(ax + b) integrates to −(1/a) cos(ax + b). cos(ax + b) integrates to (1/a) sin(ax + b).
sec²(ax + b) integrates to (1/a) tan(ax + b).
In every case: integrate as if the bracket were x, then divide by a, the number multiplying x. Add c.
This only works when the inside is linear. It does not work for something like (x² + 1)⁴.

Worked example

Integrate (2x + 1)⁴ with respect to x.

As if the bracket were x: (2x + 1)⁵ ÷ 5.
Divide by the 2 that multiplies x: (2x + 1)⁵ ÷ 10.
The answer is (2x + 1)⁵/10 + c.

Worked example

Integrate 6/(3x − 2) + sin 4x with respect to x.

6/(3x − 2) becomes 6 × (1/3) ln(3x − 2) = 2 ln(3x − 2).
sin 4x becomes −(1/4) cos 4x.
The answer is 2 ln(3x − 2) − (1/4) cos 4x + c.

Tips and tricks

Signs: sin integrates to minus cos, and cos integrates to plus sin. It is the opposite way round from differentiating.
Differentiating multiplies by a. Integrating divides by a. Check by differentiating your answer.

6 questions, about 2 minutes.

It lands in your notebook with its questions as flashcards.