Question

The area bounded by the curve y = f (x), x-axis, and the ordinates x = 1 and x = b is
(b −1) sin (3b + 4). Then, f (x) is
(a) (x − 1) cos (3x + 4)
(b) sin (3x + 4)
(c) sin (3x + 4) + 3 (x − 1) cos (3x +4)
(d) none of these

Answer 1

(c) sin (3x + 4) + 3 (x − 1) cos (3x +4)

$$\begin{gathered} y=f\left(x\right) \\ \mathrm{If}A\mathrm{is}\mathrm{the}\mathrm{the}\mathrm{area}\mathrm{bound}\mathrm{by}\mathrm{the}\mathrm{curve},x\mathit{-}\mathrm{axis},x=1\mathrm{and}x=b \\ \Rightarrow {\int }_1^{\mathrm{b}}f\left(x\right)dx={\left[A\right]}_1^b=\left(b-1\right)\sin \left(3b+4\right)\left\{\mathrm{Given}\right\} \\ \Rightarrow f\left(x\right)=\frac{d}{dx}\left(\left(x-1\right)\sin \left(3x+4\right)\right) \\ =\sin \left(3x+4\right)\frac{d}{dx}\left(x-1\right)+\left(x-1\right)\frac{d}{dx}\sin \left(3x+4\right) \\ =\sin \left(3x+4\right)+3\left(x-1\right)\cos \left(3x+4\right) \end{gathered}$$