Question

If the integral $Ι=\displaystyle \int e^{\sin x}\left(\cos ⁡ x \cdot x^{2} + 2 x\right)dx=$ $e^{f \left(x\right)}g\left(x\right)+C$ (where, $C$ is the constant of integration), then the number of solution(s) of $f\left(x\right)=g\left(x\right)$ is/are

• A. $0$
• B. $2$
• C. $4$
• D. $6$

Answer 1

Answer: (B)

As, $I=\int e^{f(x)}\left(f^{\prime}(x) g(x)+g^{\prime}(x)\right) d x=e^{f(x)} g^{(x)}+C$
Thus, $\int e^{\sin x}\left(\cos x \cdot x^{2}+2 x\right) d x=e^{\sin x} \cdot x^{2}+C$
i.e. $f(x)=\sin x \& g(x)=x^{2}$ Thus, $\displaystyle \int e^{sin x} \left(cos ⁡ x \cdot x^{2} + 2 x\right) d x = e^{sin ⁡ x} \cdot x^{2}+C$
i.e. $f\left(x\right)=sin x$ &$g\left(x\right)=x^{2}$

Which intersect at $‘2’$ points.