# Integration problem

Precursor
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 $$y$$.

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$$\int \frac{sec^2\theta}{\sqrt{1+tan^2 \theta}}d\theta$$