If _ t arrow 0 (∫_0^1(3 x+5)^ t d x)^ (1)/(t) = α 5 e ((8)/(5))^ (2)/(3) , then α is equal to ________

Correct answer: 0 aligned (α)/(5 e) & = ( _ t arrow 0 (1)/(t)(∫_0^1(3 x+5)^t d x-1)) & = ( _ t arrow 0 (1)/(t)( (3 x+5)^ t+1 3(t+1) )_0^1-1) & = ( _ t arrow 0 (1)/(t)( 8^ t+1 -5^ t+1 3(t+1) -1)) & = ( _ t arrow 0 (1)/(t)( 8^ t+1 -5^ t+1 -3 t-3 3(t+1) )) & = ( _ t arrow 0 ( 8^ t+1 · 8-5^ t+1 5-3 3(t+1) )) & = (( 8^8- 5^5-3)/(5)) aligned aligned & =((8)/(5))^ 2 / 3 (α)/(5 e)= (( ((8^8)/(5^5)))/(5)-1) & ((8)/(5))^ 2 / 3 (α)/(5)=((8^8)/(5^5))^ 1 / 3 =((8^6 · 8^2)/(5^3 · 5^2))^ 1 / 3 =(64)/(5)((8)/(5))^ 2 / 3 & α=64 aligned

JEE Main 202529 JanuaryEvening Shift

Exam: JEE Main
Year: 2025
Date: 29 January
Shift: Evening Shift
Subject: Mathematics
Chapter: Limits

JEE Main 2025 · 29 January · Evening Shift · Mathematics · Limits

If _ t arrow 0 (∫_0^1(3 x+5)^ t d x)^ (1)/(t) = α 5 e ((8)/(5))^...

\lim _{\mathrm{t} \rightarrow 0}\left(\int_0^1(3 x+5)^{\mathrm{t}} \mathrm{d} x\right)^{\frac{1}{t}}=\frac{\alpha}{5 \mathrm{e}}\left(\frac{8}{5}\right)^{\frac{2}{3}}

, then

\alpha

is equal to ________

Correct answer

Step-by-step solution

\begin{aligned} \frac{\alpha}{5 e} & =\exp \left(\lim _{t \rightarrow 0} \frac{1}{t}\left(\int_0^1(3 x+5)^t d x-1\right)\right) \\ & =\exp \left(\lim _{t \rightarrow 0} \frac{1}{t}\left(\frac{(3 x+5)^{t+1}}{3(t+1)}\right)_0^1-1\right) \\ & =\exp \left(\lim _{t \rightarrow 0} \frac{1}{t}\left(\frac{8^{t+1}-5^{t+1}}{3(t+1)}-1\right)\right) \\ & =\exp \left(\lim _{t \rightarrow 0} \frac{1}{t}\left(\frac{8^{t+1}-5^{t+1}-3 t-3}{3(t+1)}\right)\right) \\ & =\exp \left(\lim _{t \rightarrow 0}\left(\frac{8^{t+1} \cdot \ln 8-5^{t+1} \ln 5-3}{3(t+1)}\right)\right) \\ & =\exp \left(\frac{\ln 8^8-\ln 5^5-3}{5}\right)\end{aligned}

\begin{aligned} & =\left(\frac{8}{5}\right)^{2 / 3} \frac{\alpha}{5 e}=\exp \left(\frac{\ln \left(\frac{8^8}{5^5}\right)}{5}-1\right) \\ & \Rightarrow\left(\frac{8}{5}\right)^{2 / 3} \frac{\alpha}{5}=\left(\frac{8^8}{5^5}\right)^{1 / 3}=\left(\frac{8^6 \cdot 8^2}{5^3 \cdot 5^2}\right)^{1 / 3}=\frac{64}{5}\left(\frac{8}{5}\right)^{2 / 3} \\ & \Rightarrow \alpha=64\end{aligned}

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