Question

The integral $I=\int\left(\sin \left(x^{2}\right)+2 x^{2} \cos \left(x^{2}\right)\right) d x=x H(x)+C,$ (where $C$ is the constant of integration). If the range of $H\left(\right.x\left.\right)$ is $\left[a , b\right]$ , then the value of $a+2b$ is equal to

Answer 1

Given integral $I=\int\left(1 \cdot \sin \left(x^{2}\right)+x \cdot 2 x \cos \left(x^{2}\right)\right) d x$
$\left(\because \int\left(f^{\prime} g+f \cdot g^{\prime}\right) d x=\int(f g)^{\prime} d x=f \cdot g+C\right)$
$=x \cdot \sin \left(x^{2}\right)+C$
$\therefore H(x)=\sin \left(x^{2}\right),$ whose range is [-1,1]
$\therefore a=-1, b=1 \Rightarrow a+2 b=1$