Advertisements
Advertisements
Question
\[\int\frac{1 - \cos x}{1 + \cos x} dx\]
Sum
Solution
\[\int\left( \frac{1 - \cos x}{1 + \cos x} \right) dx\]
`= ∫ ( {2 sin ^2 x/2 }/ {2 cos ^2 x/2})` dx ` [ 1 - cos x = 2 sin ^2 x/2 & 1 + cos x = 2 cos ^2 x/2]`
\[ = \int \tan^2 \frac{x}{2} dx\]
\[ = \int\left( \sec^2 \frac{x}{2} - 1 \right) dx\]
\[ = \frac{\tan \frac{x}{2}}{\frac{1}{2}} - x + C\]
\[ = 2 \tan \frac{x}{2} - x + C\]
shaalaa.com
Is there an error in this question or solution?
APPEARS IN
RELATED QUESTIONS
\[\int\left( \frac{m}{x} + \frac{x}{m} + m^x + x^m + mx \right) dx\]
\[\int\frac{x^6 + 1}{x^2 + 1} dx\]
\[\int\frac{x^{- 1/3} + \sqrt{x} + 2}{\sqrt[3]{x}} dx\]
\[\int\frac{\left( x + 1 \right)\left( x - 2 \right)}{\sqrt{x}} dx\]
\[\int\frac{1}{1 - \sin x} dx\]
\[\int\frac{1}{1 + \cos 2x} dx\]
\[\int\frac{1}{1 - \cos 2x} dx\]
If f' (x) = 8x3 − 2x, f(2) = 8, find f(x)
\[\int\frac{2x + 3}{\left( x - 1 \right)^2} dx\]
\[\int\frac{2 - 3x}{\sqrt{1 + 3x}} dx\]
\[\int\frac{\sec^2 x}{\tan x + 2} dx\]
` = ∫ root (3){ cos^2 x} sin x dx `
\[\int\frac{1}{1 + \sqrt{x}} dx\]
\[\ ∫ x \text{ e}^{x^2} dx\]
\[\int\sqrt {e^x- 1} \text{dx}\]
\[\int\frac{\sin^5 x}{\cos^4 x} \text{ dx }\]
\[\int \sin^4 x \cos^3 x \text{ dx }\]
\[\int \cos^5 x \text{ dx }\]
\[\int\frac{1}{2 x^2 - x - 1} dx\]
\[\int\frac{\left( 3\sin x - 2 \right)\cos x}{13 - \cos^2 x - 7\sin x}dx\]
\[\int\frac{x^2 + x + 1}{x^2 - x + 1} \text{ dx }\]
\[\int\frac{x + 1}{\sqrt{x^2 + 1}} dx\]
\[\int x \cos^2 x\ dx\]
` ∫ sin x log (\text{ cos x ) } dx `
\[\int\left( e^\text{log x} + \sin x \right) \text{ cos x dx }\]
\[\int\frac{x^2 + x - 1}{x^2 + x - 6} dx\]
\[\int\frac{x^3}{\left( x - 1 \right) \left( x - 2 \right) \left( x - 3 \right)} dx\]
\[\int\frac{5 x^2 + 20x + 6}{x^3 + 2 x^2 + x} dx\]
\[\int\frac{dx}{\left( x^2 + 1 \right) \left( x^2 + 4 \right)}\]
\[\int\frac{x}{\left( x - 3 \right) \sqrt{x + 1}} \text{ dx}\]
If \[\int\frac{\sin^8 x - \cos^8 x}{1 - 2 \sin^2 x \cos^2 x} dx\]
\[\int\frac{\sin x + \cos x}{\sqrt{\sin 2x}} \text{ dx}\]
\[\int \tan^4 x\ dx\]
\[\int \cot^5 x\ dx\]
\[\int\frac{1}{1 + 2 \cos x} \text{ dx }\]
\[\int\frac{1 + \sin x}{\sin x \left( 1 + \cos x \right)} \text{ dx }\]
\[\int\frac{1}{x \sqrt{1 + x^n}} \text{ dx}\]
\[\int\frac{x^5}{\sqrt{1 + x^3}} \text{ dx }\]
