Q114 / 160 · AP EAMCET 2023 (16 May Shift 1) · Mathematics
Given that $\frac{d}{d x}\left[\int_0^{\phi(x)} f(t) d t\right]=\phi^{\prime}(f(x)) f^{\prime}(x)$. If $\int_0^{3^3} f(t) d t$
$=x^2 \sin 2 \pi x$, then the value of $f(8)$ is
- Correct answerA$\frac{2 \pi}{3}$
- B$\frac{4 \pi}{3}$
- C$\frac{\pi}{3}$
- D$\frac{\pi}{12}$
Solution
$\int_0^{x^3} f(t) d t=x^2 \sin 2 \pi x$
Differentiating both sides, we get
$\frac{d}{d x}\left[\int_0^{x^3} f(t) d t\right]=\frac{d}{d x}\left[x^2 \sin 2 \pi x\right]$
$\begin{aligned} & \Rightarrow f\left(x^3\right) \cdot 3 x^2=2 x \cdot \sin (2 \pi x)+2 \pi x^2 \cos (2 \pi x) \\ & \Rightarrow f\left(x^3\right)=\frac{2 x}{3 x^2}[\sin (2 \pi x)+x \pi \cos (2 \pi x)] \\ & \Rightarrow f\left(x^3\right)=\frac{2}{3 x}[\sin (2 \pi x)+x \pi \cos (2 \pi x)]\end{aligned}$
Put $x=2$ :
$\begin{aligned} & \Rightarrow f(8)=\frac{2}{3 \times 2}[\sin (4 \pi)+2 \pi \cos (4 \pi)] \\ & =\frac{1}{3}[0+2 \pi \cdot 1] \Rightarrow f(8)=\frac{2 \pi}{3} .\end{aligned}$
Hence, the correct answer is option: 'A'.