CHAPTER 03 — INTEGRATION | PART 3: TRIGONOMETRIC INTEGRATION

MATHEMATICS HIP-HOP ACADEMY • A-LEVEL PURE MATHEMATICS

CHAPTER 03 — INTEGRATION

PART 3 Trigonometric Integration

In this lesson, we learn how to integrate the basic trigonometric functions, handle expressions with angles such as 3x + 1, and use identities to integrate powers and products of sine and cosine.

Learning objectives

By the end of this part, you should be able to:

• Recall standard integrals of trigonometric functions.
• Integrate tangent and explain equivalent logarithmic forms.
• Integrate trigonometric functions with a linear expression inside the angle.
• Use trigonometric identities to integrate expressions involving sin²x, cos²x and sin x cos x.
• Check an integral by differentiating your answer.

1. Standard trigonometric integrals

Integration reverses differentiation. Therefore, we use the derivatives of trigonometric functions to establish their standard integrals. Angles are measured in radians in these formulas.

Integrand	Integral
sin x	−cos x + C
cos x	sin x + C
sec²x	tan x + C
cosec²x	−cot x + C
sec x tan x	sec x + C
cosec x cot x	−cosec x + C

Remember: The constant of integration, C, must be included in every indefinite integral.

Worked Example 1 — Basic integrals

Evaluate: ∫(4 cos x − 3 sin x + 2 sec²x) dx.

= 4∫cos x dx − 3∫sin x dx + 2∫sec²x dx

= 4 sin x + 3 cos x + 2 tan x + C.

Worked Example 2 — A cosecant expression

Evaluate: ∫(5 cosec²x − 2 cosec x cot x) dx.

= 5(−cot x) − 2(−cosec x) + C

Answer: −5 cot x + 2 cosec x + C.

2. Integrating tangent

The integral of tan x is a special result. Rewrite tangent using sine and cosine, then use substitution.

∫ tan x dx = −ln|cos x| + C = ln|sec x| + C

The two forms are equivalent because ln|sec x| = ln|1/cos x| = −ln|cos x|.

Worked Example 3 — Integrating tangent

Evaluate: ∫ tan x dx.

Write tan x = sin x / cos x.

Let u = cos x, so du = −sin x dx.

∫(sin x/cos x) dx = −∫(1/u) du

= −ln|u| + C

Answer: −ln|cos x| + C.

3. Trigonometric functions with a linear angle

When the angle is ax + b, use substitution. The derivative of ax + b is the constant a, so the final answer includes a factor of 1/a.

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

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

∫ sec²(ax + b) dx = (1/a) tan(ax + b) + C

Worked Example 4 — Sine of a linear angle

Evaluate: ∫ sin(3x + 1) dx.

Let u = 3x + 1, hence du = 3 dx and dx = du/3.

= (1/3)∫sin u du

= −(1/3)cos u + C

Answer: −⅓ cos(3x + 1) + C.

Worked Example 5 — Cosine of a linear angle

Evaluate: ∫ 4 cos(2x − 5) dx.

Let u = 2x − 5, so du = 2 dx.

= 4 × (1/2)∫cos u du

= 2 sin u + C

Answer: 2 sin(2x − 5) + C.

4. Using identities to integrate powers

Some trigonometric expressions cannot be integrated directly from the standard table. First use an identity to rewrite the expression into an integrable form.

sin²x = (1 − cos 2x)/2

cos²x = (1 + cos 2x)/2

sin x cos x = ½ sin 2x

Worked Example 6 — Integrating sin²x

Evaluate: ∫ sin²x dx.

Use sin²x = (1 − cos 2x)/2.

= ½∫(1 − cos 2x) dx

= ½[x − ½ sin 2x] + C

Answer: x/2 − sin 2x/4 + C.

Worked Example 7 — Integrating cos²x

Evaluate: ∫ cos²x dx.

Use cos²x = (1 + cos 2x)/2.

= ½∫(1 + cos 2x) dx

= ½[x + ½ sin 2x] + C

Answer: x/2 + sin 2x/4 + C.

Worked Example 8 — Product of sine and cosine

Evaluate: ∫ sin x cos x dx.

Use sin x cos x = ½ sin 2x.

= ½∫sin 2x dx

= ½[−½ cos 2x] + C

Answer: −¼ cos 2x + C.

5. Checking by differentiation

A reliable way to check an indefinite integral is to differentiate the result. You should recover the original integrand.

For example, if F(x) = −⅓ cos(3x + 1), then

F′(x) = −⅓[−sin(3x + 1)] × 3

= sin(3x + 1), which is the original integrand.

6. Common mistakes to avoid

• Forgetting the factor 1/a when integrating sin(ax + b), cos(ax + b) or sec²(ax + b).
• Writing ∫sin x dx = cos x + C. The correct answer has a negative sign.
• Forgetting the absolute value in ln|cos x|.
• Trying to integrate sin²x or cos²x directly without first using a suitable identity.
• Leaving out the constant of integration.
• Using degree-based differentiation formulas without converting to radians.

7. Practice exercises

Attempt each question before opening its answer. Give answers in terms of x and include C for indefinite integrals.

1. Evaluate ∫(3 sin x + 2 cos x) dx.

Reveal answer to Question 1

−3 cos x + 2 sin x + C.

2. Evaluate ∫(4 sec²x − 5 cosec²x) dx.

Reveal answer to Question 2

4 tan x + 5 cot x + C.

3. Evaluate ∫ tan x dx.

Reveal answer to Question 3

−ln|cos x| + C, equivalently ln|sec x| + C.

4. Evaluate ∫ cos(5x) dx.

Reveal answer to Question 4

⅕ sin 5x + C.

5. Evaluate ∫ 6 sin(2x + 3) dx.

Reveal answer to Question 5

−3 cos(2x + 3) + C.

6. Evaluate ∫ sec²(4x − 1) dx.

Reveal answer to Question 6

¼ tan(4x − 1) + C.

7. Evaluate ∫ sin²x dx.

Reveal answer to Question 7

x/2 − ¼ sin 2x + C.

8. Evaluate ∫ cos²x dx.

Reveal answer to Question 8

x/2 + ¼ sin 2x + C.

9. Evaluate ∫ sin x cos x dx.

Reveal answer to Question 9

−¼ cos 2x + C. An equivalent answer is ½ sin²x + C.

10. Evaluate ∫(2 cos 3x − 4 sin 2x) dx.

Reveal answer to Question 10

⅔ sin 3x + 2 cos 2x + C.

Part 3 summary

• Know the standard trigonometric integrals and their signs.
• For a linear angle ax + b, account for the factor 1/a.
• Use identities to rewrite powers and products before integrating.
• Differentiate your answer to check it.

Next: Part 4 — Partial Fractions and Integration by Parts.

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