This document provides examples of calculating antiderivatives (indefinite integrals) of trigonometric functions. It defines the basic trigonometric functions of sine, cosine, tangent, cosecant, secant, and cotangent. It then shows the step-by-step workings for finding antiderivatives of expressions involving trigonometric functions, such as (cos x - sin x) dx, cot^2x dx, tan^2x dx, sin x / cos^2x dx, (8cos x + 3 sin x) dx, and (4sec^2x - sec x tan x) dx. The document aims to demonstrate the process of calculating antiderivatives of trigon

1. ANTIDERIVATIVES OF
TRIGONOMETRIC
FUNCTIONS
ANTIDERIVATIVES OF
TRIGONOMETRIC
FUNCTIONS
BASIC CALCULUS
MS. ELLEN NASAYAO

2. TRIGONOMETRIC FUNCTIONS
Sine θ = opposite/hypotenuse
Cosine θ = adjacent/hypotenuse
Tangent θ = opposite/adjacent
Cosecant θ = hypotenuse/opposite​
Secant θ = hypotenuse/adjacent​
​Cotangent θ = adjacent/opposite
TRIGONOMETRIC FUNCTIONS

3. TRIGONOMETRIC IDENTITIES
TRIGONOMETRIC IDENTITIES

4. ANTIDERIVATIVES OF TRIGONOMETRIC
FUNCTIONS
ANTIDERIVATIVES OF TRIGONOMETRIC
FUNCTIONS

5. EXAMPLE 1
(𝑐𝑜𝑠 𝑥 − 𝑠𝑖𝑛 𝑥) 𝑑𝑥
= 𝑐𝑜𝑠 𝑥 𝑑𝑥 − 𝑠𝑖𝑛 𝑥 𝑑𝑥
= 𝑠𝑖𝑛 𝑥 − (− 𝑐𝑜𝑠 𝑥) + 𝐶
= 𝑠𝑖𝑛 𝑥 + 𝑐𝑜𝑠 𝑥 + 𝐶

6. EXAMPLE 2
𝑐𝑜𝑡2
𝑥 𝑑𝑥
𝑐𝑜𝑡2
𝑥 + 1 = 𝑐𝑠𝑐2
𝑥
𝑐𝑜𝑡2
𝑥= 𝑐𝑠𝑐2
𝑥 − 1
= (𝑐𝑠𝑐2
𝑥 − 1)𝑑𝑥
= 𝑐𝑠𝑐2
𝑥 𝑑𝑥 − 1 𝑑𝑥
= − 𝑐𝑜𝑡 𝑥 − 𝑥 + 𝐶

7. EXAMPLE 3
𝑡𝑎𝑛2
𝑥 𝑑𝑥
𝑡𝑎𝑛2
𝑥 + 1 = 𝑠𝑒𝑐2
𝑥
𝑡𝑎𝑛2
𝑥 = 𝑠𝑒𝑐2
𝑥 − 1
= (𝑠𝑒𝑐2
𝑥 − 1)𝑑𝑥
= 𝑠𝑒𝑐2
𝑥 𝑑𝑥 − 1 𝑑𝑥
= 𝑡𝑎𝑛 𝑥 − 𝑥 + 𝐶

8. EXAMPLE 4
𝑠𝑖𝑛 𝑥
𝑐𝑜𝑠2𝑥
𝑑𝑥
= (
𝑠𝑖𝑛 𝑥
𝑐𝑜𝑠 𝑥 𝑐𝑜𝑠 𝑥
)𝑑𝑥
= (
1
𝑐𝑜𝑠 𝑥
∙
𝑠𝑖𝑛 𝑥
𝑐𝑜𝑠 𝑥
)𝑑𝑥
= (𝑠𝑒𝑐 𝑥 ∙ 𝑡𝑎𝑛 𝑥) 𝑑𝑥
= 𝑠𝑒𝑐 𝑥 + 𝐶

9. EXAMPLE 5
(8𝑐𝑜𝑠 𝑥 + 3 𝑠𝑖𝑛 𝑥) 𝑑𝑥
= 8 𝑐𝑜𝑠 𝑥 + 3 𝑠𝑖𝑛 𝑥 𝑑𝑥
= 8 𝑠𝑖𝑛 𝑥 + 3(− 𝑐𝑜𝑠 𝑥) + 𝐶
= 8 𝑠𝑖𝑛 𝑥 − 3 𝑐𝑜𝑠 𝑥 + 𝐶

10. EXAMPLE 6
(4𝑠𝑒𝑐2
𝑥 − 𝑠𝑒𝑐 𝑥 𝑡𝑎𝑛 𝑥) 𝑑𝑥
= 4 𝑠𝑒𝑐2
𝑥 − (𝑠𝑒𝑐 𝑥 𝑡𝑎𝑛 𝑥) 𝑑𝑥
= 4 𝑡𝑎𝑛 𝑥 − 𝑠𝑒𝑐 𝑥 + 𝐶

11. WHAT IF I-TRY NATIN???
1. 5 𝑐𝑠𝑐 𝑥 𝑐𝑜𝑡 𝑥 𝑑𝑥
2. 3𝑠𝑒𝑐2𝑥 𝑑𝑥