Ex 7.8, 15 - Chapter 7 Class 12 Integrals

Last updated at Dec. 16, 2024 by Teachoo

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Ex 7.8, 15 ∫_0^1▒〖𝑥𝑒^(𝑥^2 ) 〗 𝑑𝑥 Step 1 :- Let F(𝑥)=∫1▒〖𝑥𝑒^(𝑥^2 ) 𝑑𝑥〗 Let 𝑥^2=𝑡 Differentiating w.r.t. 𝑥 2𝑥=𝑑𝑡/𝑑𝑥 𝑑𝑥=𝑑𝑡/2𝑥 Therefore, ∫1▒〖𝑥 𝑒^(𝑥^2 ) 𝑑𝑥=∫1▒〖𝑥 𝑒^𝑡 𝑑𝑡/2𝑥〗〗 =1/2 ∫1▒〖𝑒^𝑡 𝑑𝑡〗 =1/2 𝑒^𝑡 Putting 𝑡=𝑥^2 =1/2 𝑒^(𝑥^2 ) Hence, F(𝑥)=1/2 𝑒^(𝑥^2 ) Step 2 :- ∫_0^1▒〖𝑥𝑒^(𝑥^2 ) 𝑑𝑥〗=𝐹(1)−𝐹(0) =1/2 𝑒^(1^2 )−1/2 𝑒^(0^2 ) =1/2 𝑒^1−1/2 𝑒^0 =𝟏/𝟐 (𝒆−𝟏)

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