Solving an Integration Problem with Homework Statement

[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 $$y$$.

 

[rock.freak667]

From lny=tanθ, you should get that dy/y =sec²θ dθ

giving you


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