First slide

Introduction to integration

Question

If $f (x) = \cos x - \cos^{2} x + \cos^{3} x - \dots \infty$ then $\int f (x) \ddot{d} x$ is equal to

Moderate

A

$\tan \frac{x}{2} + C$
B

$x - \tan \frac{x}{2} + C$
C

$x - \frac{1}{2} \tan \frac{x}{2} + C$
D

$\frac{x - \tan \frac{x}{2}}{2} + C$

Solution

$\begin{matrix} ∵ f (x) = \cos x - \cos^{2} x + \cos^{3} x - \dots \infty = \frac{\cos x}{1 + \cos x} \\ ∴ \int f (x) dx = \int \frac{1 + \cos x}{1 + \cos x} dx - \int \frac{1}{1 + \cos x} dx \\ = x - \frac{1}{2} \int \sec^{2} \frac{x}{2} dx \\ = x - \frac{1}{2} \tan \frac{x}{2} \cdot 2 + C = x - \tan \frac{x}{2} + C \end{matrix}$

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