First slide

Methods of integration

Question

$\int \frac{1}{\sin^{3} x \sin (x + α)} d x$

Moderate

A

$- 2 \cos e c α {(\cos α - \tan x \sin α)}^{\frac{1}{2}} + c$
B

$- 2 {(\cos α + c o t x \sin α)}^{\frac{1}{2}} + c$
C

$- 2 \cos e c α {(\cos α + c o t x \sin α)}^{\frac{1}{2}} + c$
D

$- 2 \cos e c α {(s i n α + c o t x \cos α)}^{\frac{1}{2}} + c$

Solution

$\begin{array}{l} I = \int \frac{1}{\sqrt{\sin^{3} x \sin (x + α)}} d x = \int \frac{1}{\sqrt{\sin^{3} x (\sin x \cos α + \sin α \cos x)}} d x \\ = \int \frac{1}{\sqrt{\sin^{4} x (\cos α + \sin α c o t x)}} d x \\ = \int \frac{1}{\sin^{2} x \sqrt{(\cos α + \sin α c o t x)}} d x \\ = \int \frac{\cos e c^{2} x}{\sqrt{(\cos α + \sin α c o t x)}} d x (\begin{array}{l} p u t (\cos α + \sin α c o t x) = t \\ \Rightarrow - \cos e c^{2} x \sin α d x = d t \end{array}) \\ = - \int \frac{1}{\sin α \sqrt{t}} d t \\ = - \frac{1}{\sin α} (\frac{t^{\frac{1}{2}}}{\frac{1}{2}}) + c \\ = - 2 \cos e c α \sqrt{t} + c \\ = - 2 \cos e c α {(\cos α + c o t x \sin α)}^{\frac{1}{2}} + c \end{array}$

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