INTEGRATION OF EXPONENTIAL
FUNCTIONS
Let u be a differentiable function of x.
∫
𝑒
𝑢
=𝑒
𝑢
+𝐶
∫
𝑎
𝑢
𝑑𝑢=𝑎
𝑢
𝑙𝑛𝑎 +𝐶
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Integrals of Exponential Functions, Lecture notes of Engineering Mathematics
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2021/2022
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INTEGRATION OF EXPONENTIAL
FUNCTIONS
Let u be a differentiable function of x.
𝑑𝑢 = 𝑎
𝑙𝑛𝑎
- 𝐶
1.∫ 𝑥 𝑒 3 𝑥 2
- 4 𝑑𝑥 (^) ¿ ∫ 𝑒 3 𝑥 2
- 4
. 𝑥 𝑑𝑥
2
- 4 𝑑𝑢 = 6 𝑥𝑑𝑥 𝑑𝑢 6
¿∫ 𝑒 𝑢 𝑑𝑢 6 . ¿ 1 6 ∫ 𝑒 𝑢 𝑑𝑢 ¿
𝑢
- 𝐶 ¿
3 𝑥^2 + 4
- 𝐶 2.∫ 𝑒 1 𝑥 𝑥 2 𝑑𝑥 (^) ¿∫ 𝑒 1 𝑥 𝑑𝑥 𝑥 2
¿ ∫ 𝑒 𝑢 ( − 𝑑𝑢 ) ¿ −𝑒 1 𝑥
- 𝐶 ∫ 𝑥^ 𝑒 3 𝑥 2
- 4 𝑑𝑥 ∫ 𝑒 1 𝑥 𝑥 2 𝑑𝑥
Practice: 1.∫ (^ 𝑒 2 𝑥
- 𝑒 −𝑥 ) 2 𝑑𝑥 2.∫ 𝑒 2 𝑥 − 1 𝑒 𝑥 𝑑𝑥 3.∫ 5 𝑒 𝑙𝑛𝑥 2 𝑥 𝑑𝑥 𝐸𝑣𝑎𝑙𝑢𝑎𝑡𝑒 𝑡 𝑒h 𝑓𝑜𝑙𝑙𝑜𝑤𝑖𝑛𝑔 𝑖𝑛𝑡𝑒𝑔𝑟𝑎𝑙 :