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Homework Statement
Use appropriate substitution and than a trigonometric substitution and evaluate the integral.
\int_{1}^{e}\frac{dy}{y\sqrt{1 + (lny)^{2}}}
The attempt at a solution
\int_{1}^{e}\frac{dy}{y\sqrt{1 + (lny)^{2}}}
ln y = tan\theta
y = cos^{2}\theta
dy = -2cos\theta sin\theta d\theta
= -2\int_{1}^{e}\frac{cos\theta sin\theta d\theta}{cos^{2}\theta\sqrt{1 + tan^{2}\theta}}
= -2\int_{1}^{e}\frac{sin\theta d\theta}{cos\theta sec\theta}
= -2\int_{1}^{e}sin\theta d\theta
How do I proceed from here? I think I have to change the limits of integration in terms of \theta instead of y.
Use appropriate substitution and than a trigonometric substitution and evaluate the integral.
\int_{1}^{e}\frac{dy}{y\sqrt{1 + (lny)^{2}}}
The attempt at a solution
\int_{1}^{e}\frac{dy}{y\sqrt{1 + (lny)^{2}}}
ln y = tan\theta
y = cos^{2}\theta
dy = -2cos\theta sin\theta d\theta
= -2\int_{1}^{e}\frac{cos\theta sin\theta d\theta}{cos^{2}\theta\sqrt{1 + tan^{2}\theta}}
= -2\int_{1}^{e}\frac{sin\theta d\theta}{cos\theta sec\theta}
= -2\int_{1}^{e}sin\theta d\theta
How do I proceed from here? I think I have to change the limits of integration in terms of \theta instead of y.
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