Question

Integration of tan^-1(cosx/1+sinx)

Answer 1

(π/4)x - (x^2)/4 + C

Answer 2

To integrate the given expression, we first simplify the argument of the $\tan^{-1}$ function.

The expression inside $\tan^{-1}$ is $\frac{\cos x}{1+\sin x}$.

We can use trigonometric identities to simplify this expression. Recall the identities:

1. $\cos x = \sin(\frac{\pi}{2} - x)$
2. $\sin x = \cos(\frac{\pi}{2} - x)$

Substitute these into the expression: $\frac{\cos x}{1+\sin x} = \frac{\sin(\frac{\pi}{2} - x)}{1+\cos(\frac{\pi}{2} - x)}$ Let $A = \frac{\pi}{2} - x$. The expression becomes $\frac{\sin A}{1+\cos A}$.

Now, use the half-angle formulas for $\sin A$ and $1+\cos A$:

1. $\sin A = 2\sin\left(\frac{A}{2}\right)\cos\left(\frac{A}{2}\right)$
2. $1+\cos A = 2\cos^2\left(\frac{A}{2}\right)$

Substitute these into the expression: $\frac{\sin A}{1+\cos A} = \frac{2\sin\left(\frac{A}{2}\right)\cos\left(\frac{A}{2}\right)}{2\cos^2\left(\frac{A}{2}\right)} = \frac{\sin\left(\frac{A}{2}\right)}{\cos\left(\frac{A}{2}\right)} = \tan\left(\frac{A}{2}\right)$ Now, substitute back $A = \frac{\pi}{2} - x$: $\tan\left(\frac{A}{2}\right) = \tan\left(\frac{\frac{\pi}{2} - x}{2}\right) = \tan\left(\frac{\pi}{4} - \frac{x}{2}\right)$ So, the original integral becomes: $\int \tan^{-1}\left(\tan\left(\frac{\pi}{4} - \frac{x}{2}\right)\right) dx$ For the principal value of $\tan^{-1}$, if $-\frac{\pi}{2} < \theta < \frac{\pi}{2}$, then $\tan^{-1}(\tan \theta) = \theta$. Assuming $x$ is in a range such that $\frac{\pi}{4} - \frac{x}{2}$ lies within $(-\frac{\pi}{2}, \frac{\pi}{2})$, we can simplify the integrand: $\tan^{-1}\left(\tan\left(\frac{\pi}{4} - \frac{x}{2}\right)\right) = \frac{\pi}{4} - \frac{x}{2}$ Now, we need to integrate this simplified expression: $\int \left(\frac{\pi}{4} - \frac{x}{2}\right) dx$ This is a straightforward integration: $\int \frac{\pi}{4} dx - \int \frac{x}{2} dx$ $= \frac{\pi}{4}x - \frac{1}{2} \int x dx$ $= \frac{\pi}{4}x - \frac{1}{2} \left(\frac{x^2}{2}\right) + C$ $= \frac{\pi}{4}x - \frac{x^2}{4} + C$ where $C$ is the constant of integration.