Gujarat BoardEnglish MediumSTD 12 ScienceMathsIntegrals2 Marks
Question
Integrate the rational function $\frac{\cos x}{(1-\sin x)(2-\sin x)}$ [Hint: Put sin x = t]
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Answer
Given function is $\frac{\cos x}{(1-\sin x)(2-\sin x)}$ Let sin x = t Therefore, $\int \frac{\cos x}{(1-\sin x)(2-\sin x)} d x=\int \frac{d t}{(1-t)(2-t)}$ Let $\frac{1}{(1-t)(2-t)}=\frac{A}{(1-t)}+\frac{B}{(2-t)}$ $\Rightarrow$ 1 = A(2 - t) + B (1 - t) …(i) Substituting t = 2 and then t = 1 in equation (i), we get, Therefore, $\frac{1}{(1-t)(2-t)}=\frac{1}{(1-t)}-\frac{1}{(2-t)}$ $\Rightarrow$$\int \frac{\cos x}{(1-\sin x)(2-\sin x)} d x=\int\left\{\frac{1}{(1-t)}-\frac{1}{(2-t)}\right\} d t$ = $-log |1-t|+\log |2-t|+C$ = $\log \left|\frac{2-t}{1-t}\right|+C$ = $\log \left|\frac{2-\sin x}{1-\sin x}\right|+C$
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