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