Integral Calculus
Indefinite Integrals
MMTS_Full_Test_18
Grade 12
Question:
If $\int\dfrac{5\tan x\,dx}{\tan x-2}=x+a\ln|\sin x-2\cos x|+C$, then $a$ is equal to ($C$ is constant of integration)
Step-by-Step Solution
Key Concept: Write $5\tan x = A(\tan x-2)+B\sec^2 x$... or $5\sin x=A(\sin x-2\cos x)+B(\cos x+2\sin x)$
$5\sin x=A(\sin x-2\cos x)+B\cos x+2B\sin x$. $A+2B=5$, $-2A+B=0\Rightarrow B=2A\Rightarrow 5A=5\Rightarrow A=1,B=2$. $\int\frac{5\sin x}{\sin x-2\cos x}dx=x+2\ln|\sin x-2\cos x|+C$. $a=2$.
Correct Answer: 2