Question

If linear function $f(x)$ and $g(x)$ satisfy $\int [(3x-1)\cos x+(1-2x)\sin x]dx=f(x)\cos x+g(x)\sin x+C,$
then

• A. $f(x)=3x-5$
• B. $g(x)=3+x$
• C. $g(x)=3(x-1)$
• D. $f(x)=3(x-1)$

Answer 1

Answer: (C)

Given, $f(x)\cos x+g(x)\sin x+C$
$=\int [(3x-1)\cos x+(1-2x)\sin x]dx$
$=\int \underset{I}{(3x-1)}\underset{II}{\mathrm{cosxdx}}+\int \underset{I}{(1-2x)}\underset{II}{\mathrm{sinxdx}}$
$=(3x-1)\sin x-\int 3\sin xdx-(1-2x)\cos x+\int (-2)\cos xdx$
(using integration by parts)
$=(3x-1)\sin x+3\cos x-(1-2x)\cos x-2\sin x+C$
$=(3x-1-2)\sin x+(3-1+2x)\cos x+C$
$\Rightarrow f(x)\cos x+g(x)\sin x+C$
$=3(x-1)\sin x+2(x+1)\cos x+C$
On comparing the coefficients of $\cos x$ and $\sin x$, we get
$f(x)=2(x+1)$ and $g(x)=3(x-1)$