Question

If the integral $I=\int e^{\sin x}\left(\cos x\cdot x^2+2x\right)dx=$ $e^{f\left(x\right)}g\left(x\right)+C$ (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, $I=\int e^{f(x)}\left(f^{\prime }(x)g(x)+g^{\prime }(x)\right)dx=e^{f(x)}{g}^{(x)}+C$
Thus, $\int e^{\sin x}\left(\cos x\cdot x^2+2x\right)dx=e^{\sin x}\cdot x^2+C$
i.e. $f(x)=\sin x\&g(x)=x^2$ Thus, $\int e^{sinx}\left(cosx\cdot x^2+2x\right)dx=e^{sinx}\cdot x^2+C$
i.e. $f\left(x\right)=sinx$ &$g\left(x\right)=x^2$

Which intersect at $‘2’$ points.