Every standard derivative gives a standard integral. The key results are ∫ eˣ dx = eˣ + c, ∫ 1/x dx = ln|x| + c, ∫ sin x dx = −cos x + c, ∫ cos x dx = sin x + c and ∫ sec²x dx = tan x + c, with x in radians. Integrate sums term by term, and keep definite integral answers exact, in terms of e, ln or surds.
Standard integrals
- ∫ eˣ dx = eˣ + c
- ∫ 1/x dx = ln|x| + c
- ∫ sin x dx = −cos x + c
- ∫ cos x dx = sin x + c
- ∫ sec²x dx = tan x + c
- ∫ sec x tan x dx = sec x + c
- ∫ cosec²x dx = −cot x + c
- ∫ cosec x cot x dx = −cosec x + c
Integrating eˣ and 1/x
eˣ is its own integral, just as it is its own derivative. The integral of 1/x is ln|x| + c: this is the case the power rule cannot handle, because n = −1 would mean dividing by zero. The modulus lets the result work for negative x as well. Constant multiples pass straight through, so ∫ 5/x dx = 5 ln|x| + c.
The basic trig integrals
Reverse the standard derivatives: sin x integrates to −cos x, cos x integrates to sin x, and sec²x integrates to tan x. The sign changes only for sine. Work in radians, especially for definite integrals with limits such as π/2.
The wider set of trig integrals
Reversing the derivatives of sec, cosec and cot gives three more results: sec x tan x integrates to sec x, cosec²x integrates to −cot x, and cosec x cot x integrates to −cosec x. Two of these carry a minus sign, which is where most errors happen, so check by differentiating your answer.
Integrating combinations term by term
A sum of standard functions is integrated one term at a time, keeping each constant multiple. Mixing eˣ, trig and 1/x in a single expression is exactly where sign and result slips happen, so take each term on its own.
Checking by differentiating
Every standard integral can be checked by differentiating the answer. This is especially useful for the reciprocal trig results, where the signs are easy to confuse: if differentiating your answer does not give back the integrand exactly, including its sign, adjust it before moving on.
Definite integrals and exact answers
Evaluate between limits as usual, by substituting the upper and lower limits and subtracting. Answers with standard functions are usually exact expressions such as e − 1, ln 5 or √2 − 1, and questions expect them in that form rather than as decimals. Useful facts: e^(ln a) = a, ln 1 = 0, sin 0 = 0 and cos 0 = 1.
Worked examples
Example 1. Find ∫ (2eˣ + 1/x) dx.
- 2eˣ integrates to 2eˣ.
- 1/x integrates to ln|x|.
Answer: 2eˣ + ln|x| + c.
Marks: M1 for 2eˣ, A1 for ln|x| + c.
Example 2. Find ∫ 3 cos x dx.
- ∫ cos x dx = sin x.
- Keep the multiple 3.
Answer: 3 sin x + c.
Marks: A1 for 3 sin x + c.
Example 3. Find ∫ cosec²x dx.
- The derivative of cot x is −cosec²x.
- Reverse it, keeping track of the sign.
Answer: −cot x + c.
Marks: A1 for −cot x + c.
Example 4. Find ∫ (eˣ + sin x) dx.
- eˣ integrates to eˣ.
- sin x integrates to −cos x.
Answer: eˣ − cos x + c.
Marks: A1 for eˣ − cos x + c.
Example 5. Find the exact value of ∫ (2/x + cos x) dx from π/2 to π.
- Integrate: [2 ln x + sin x] from π/2 to π.
- Upper limit: 2 ln π + sin π = 2 ln π.
- Lower limit: 2 ln(π/2) + sin(π/2) = 2 ln(π/2) + 1.
- Subtract: 2 ln π − 2 ln(π/2) − 1 = 2 ln 2 − 1.
Answer: 2 ln 2 − 1.
Marks: M1 for the integral, M1 for substituting the limits, A1 for 2 ln 2 − 1.
Common mistakes
- Using the power rule on 1/x. ∫ x⁻¹ dx is ln|x| + c, not x⁰/0.
- Getting the sign wrong for sin. ∫ sin x dx = −cos x + c; ∫ cos x dx = +sin x + c.
- Forgetting the minus signs on cosec²x and cosec x cot x. Their integrals are −cot x and −cosec x.
- Rounding exact answers. Leave answers such as e − 1 or 2 ln 2 − 1 exact unless told otherwise.
- Using degrees with trig limits. Integration with trig functions is always done in radians.
How standard integrals are examined
In Edexcel International A-Level, standard integrals are Pure 3 content (WMA13). In Edexcel A-Level (9MA0) they are covered in the second year of Pure. Questions ask you to:
- integrate a combination of eˣ, 1/x and trig terms
- find the exact area under a curve such as y = eˣ
- find a limit that gives a stated area
- find the area between two standard curves, such as sin x and cos x
You can practise these with full mark schemes in the Pure 3 integration exam questions. The Pure 3 integration worksheet and the A-Level integration revision engine provide more.
Each result reverses one of the standard derivatives. Next, integration of f(ax + b) extends them to linear arguments.
Frequently asked questions
What is the integral of 1/x?
ln|x| + c. The modulus allows the result to hold for negative values of x.
What is the integral of sin x?
−cos x + c. The minus sign appears because the derivative of cos x is −sin x.
Why must trig integrals use radians?
The derivative of sin x is cos x only in radians, so the integrals built from those derivatives only hold in radians.