ANTIDERIVATIVES OF TRIGONOMETRIC FUNCTIONS.pptx

ANTIDERIVATIVES OF
TRIGONOMETRIC
FUNCTIONS
ANTIDERIVATIVES OF
TRIGONOMETRIC
FUNCTIONS
BASIC CALCULUS
MS. ELLEN NASAYAO
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
TRIGONOMETRIC IDENTITIES
TRIGONOMETRIC IDENTITIES
ANTIDERIVATIVES OF TRIGONOMETRIC
FUNCTIONS
ANTIDERIVATIVES OF TRIGONOMETRIC
FUNCTIONS
EXAMPLE 1
(𝑐𝑜𝑠 𝑥 − 𝑠𝑖𝑛 𝑥) 𝑑𝑥
= 𝑐𝑜𝑠 𝑥 𝑑𝑥 − 𝑠𝑖𝑛 𝑥 𝑑𝑥
= 𝑠𝑖𝑛 𝑥 − (− 𝑐𝑜𝑠 𝑥) + 𝐶
= 𝑠𝑖𝑛 𝑥 + 𝑐𝑜𝑠 𝑥 + 𝐶
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
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
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
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
EXAMPLE 6
(4𝑠𝑒𝑐2
𝑥 − 𝑠𝑒𝑐 𝑥 𝑡𝑎𝑛 𝑥) 𝑑𝑥
= 4 𝑠𝑒𝑐2
𝑥 − (𝑠𝑒𝑐 𝑥 𝑡𝑎𝑛 𝑥) 𝑑𝑥
= 4 𝑡𝑎𝑛 𝑥 − 𝑠𝑒𝑐 𝑥 + 𝐶
WHAT IF I-TRY NATIN???
1. 5 𝑐𝑠𝑐 𝑥 𝑐𝑜𝑡 𝑥 𝑑𝑥
2. 3𝑠𝑒𝑐2𝑥 𝑑𝑥
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ANTIDERIVATIVES OF TRIGONOMETRIC FUNCTIONS.pptx