Question

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

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

Answer 1

Answer: (D)

$g\left(x\right)=3(x-1)$

$\int \left[\right(3x-1)\cos x+(1-2x\left)\sin x\right]dx$

$=\int \left[x\right(3\cos x-2\sin x)+\sin x-\cos x]dx$

$=\int x(3\cos x-2\sin x)dx+\int (\sin x-\cos x)dx$

$=x(3\sin x+2\cos x)-\int (3\sin x+2\cos x)dx-\cos x-\sin x$

$=x(3\sin x+2\cos x)-(2\sin x-3\cos x)-\cos x-\sin x+C$

$=x(3\sin x+2\cos x)+2\cos x-3\sin x+C$

$=\cos x(2x+2)+\sin x(3x-3)+C$

Hence $f\left(x\right)=2x+2$ and $g\left(x\right)=3x-3$.