Question

The value of the integral $Ι=\int e^{x}\left(sin x + cos ⁡ x\right)dx$ is equal to $e^{x}\cdot f\left(x\right)+C,$ $C$ being the constant of integration. Then the maximum value of $y=f\left(x^{2}\right),\forall x\in R$ is equal to

• A. $0$
• B. $-1$
• C. $1$
• D. $\frac{1}{2}$

Answer 1

Answer: (C)

As we know,
$\int e^{x}\left(f(x)+f^{\prime}(x)\right) d x=e^{x} \cdot f(x)+C$
Thus, $\int e^{x} \left(sin x + cos ⁡ x\right) d x = e^{x} \cdot sin ⁡ x + C$
i.e. $f\left(x\right)=sin x$
Hence, $f\left(x^{2}\right)=sin\left(x^{2}\right):$ which has the maximum value of $‘1’$