Question

Solve the following and elaborate all steps

$\int \: \frac{sinx}{sin8x \: sin7x} \: dx$
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Answer 1

$\large\underline{\sf{Solution-}}$

Given integral is

$\displaystyle\int\rm \frac{sinx}{sin8x \: sin7x} \: dx$

can be rewritten as

$\rm \: = \: \displaystyle\int\rm \frac{sin(8x - 7x)}{sin8x \: sin7x} \: dx$

We know,

$\boxed{ \rm{ \:sin(x - y) = sinx \: cosy \: - \: siny \: cosx \: \: }}$

So, using this result, we get

$\rm \: = \: \displaystyle\int\rm \frac{sin8x \: cos7x - sin7x \: cos8x}{sin8x \: sin7x} \: dx$

$\rm \: = \: \displaystyle\int\rm \frac{sin8x \: cos7x}{sin8x \: sin7x} \: dx - \displaystyle\int\rm \frac{sin7x \: cos8x}{sin8x \: sin7x} \: dx$

$\rm \: = \displaystyle\int\rm cot7x \: dx \: - \: \displaystyle\int\rm cot8x \: dx$

We know,

$\boxed{ \rm{ \:\displaystyle\int\rm cotx \: dx \: = \: log |sinx| + c \: }}$

So, using this result, we get

$\rm \: = \dfrac{log |sin7x| }{7} - \dfrac{log |sin8x| }{8} + c$

Hence,

$\rm \:\displaystyle\int\rm \frac{sinx}{sin8x \: sin7x}dx = \dfrac{log |sin7x| }{7} - \dfrac{log |sin8x| }{8} + c$

$\rule{190pt}{2pt}$

Additional Information :-

$\begin{gathered}\begin{gathered}\boxed{\begin{array}{c|c} \bf f(x) & \bf \displaystyle \int \rm \:f(x) \: dx\\ \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf k & \sf kx + c \\ \\ \sf sinx & \sf - \: cosx+ c \\ \\ \sf cosx & \sf \: sinx + c\\ \\ \sf {sec}^{2} x & \sf tanx + c\\ \\ \sf {cosec}^{2}x & \sf - cotx+ c \\ \\ \sf secx \: tanx & \sf secx + c\\ \\ \sf cosecx \: cotx& \sf - \: cosecx + c\\ \\ \sf tanx & \sf logsecx + c\\ \\ \sf \dfrac{1}{x} & \sf logx+ c\\ \\ \sf {e}^{x} & \sf {e}^{x} + c\end{array}} \\ \end{gathered}\end{gathered}$