Chapter 7 Integrals

Notes

Integrals

1. Integral Calculus (Anti-Derivatives)

The inverse process of differentiation is known as integration.

If:

d/dx {F(x)} = f(x)

then:

∫ f(x) dx = F(x) + C

where C is called the constant of integration.

2. Some Useful Formulae

• ∫ cos x dx = sin x + C
• ∫ sin x dx = -cos x + C
• ∫ sec²x dx = tan x + C
• ∫ cosec²x dx = -cot x + C
• ∫ sec x tan x dx = sec x + C
• ∫ cosec x cot x dx = -cosec x + C
• ∫ xⁿ dx = xⁿ⁺¹/(n + 1) + C, n ≠ -1
• ∫ (1/x) dx = log|x| + C
• ∫ (1/x²) dx = -1/x + C
• ∫ aˣ dx = aˣ/log a + C
• ∫ eˣ dx = eˣ + C
• ∫ c dx = cx + C
• ∫ dx/√x = 2√x + C
• ∫ dx/√(1 - x²) = sin⁻¹x + C
• ∫ dx/(1 + x²) = tan⁻¹x + C
• ∫ dx/(|x|√(x² - 1)) = sec⁻¹x + C

3. Important Trigonometric Integrals

3.1 Integral of sin²x

Using:

sin²x = (1 - cos 2x)/2

Therefore,

∫ sin²x dx = x/2 - (sin 2x)/4 + C

3.2 Integral of cos²x

Using:

cos²x = (1 + cos 2x)/2

Therefore,

∫ cos²x dx = x/2 + (sin 2x)/4 + C

3.3 Integral of tan²x

Using:

tan²x = sec²x - 1

Therefore,

∫ tan²x dx = tan x - x + C

3.4 Integral of cot²x

Using:

cot²x = cosec²x - 1

Therefore,

∫ cot²x dx = -cot x - x + C

3.5 Integral of tan x

tan x = sin x/cos x

Therefore,

∫ tan x dx = -log|cos x| + C

or,

∫ tan x dx = log|sec x| + C

3.6 Integral of cot x

cot x = cos x/sin x

Therefore,

∫ cot x dx = log|sin x| + C

3.7 Integral of sec x

∫ sec x dx = log|sec x + tan x| + C

Also,

∫ sec x dx = log|tan(π/4 + x/2)| + C

3.8 Integral of cosec x

∫ cosec x dx = log|cosec x - cot x| + C

Also,

∫ cosec x dx = log|tan(x/2)| + C

4. Finding an Anti-Derivative with a Given Condition

If a condition such as F(a) = b is given, first integrate the given derivative and then use the condition to find the constant of integration.

Example

Find F(x) if:

F'(x) = 4x³ - 6