Question

(b) What is the integral of 2x cosec^x^dx
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Answer 1

Answer:

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Step-by-step explanation:

Notice,

Method 1:

[math]\int cosec x\ dx=\int \frac{cosec x(cosec x-\cot x)}{(cosec x-\cot x)}\ dx[/math]

[math]=\int \frac{d(cosec x-\cot x)}{(cosec x-\cot x)}[/math]

[math]=\ln|cosec x-\cot x|+C[/math]

[math]=-\ln|cosec x+\cot x|+C[/math]

Method 2:

[math]\int cosec x\ dx=\int \frac{1}{\sin x}\ dx[/math]

[math]=\int \frac{1}{\frac{2\tan\frac x2}{1+\tan^2\frac x2}}\ dx[/math]

[math]=\int \frac{1+\tan^2\frac x2}{2\tan\frac x2}\ dx[/math]

[math]=\int \frac{\frac12\sec^2\frac x2}{\tan\frac x2}\ dx[/math]

[math]=\int \frac{d\left(\tan\frac x2\right)}{\tan\frac x2}[/math]

[math]=\ln\left|\tan\frac{x}{2}\right|+C[/math]

[math]\therefore \boxed{\int cosec x\ dx=\ln|cosec x-\cot x|=-\ln|cosec x+\cot x|=\ln\left|\tan\frac x2\right|}[/math]