Q109 / 160 · AP EAMCET 2024 (19 May Shift 2) · Mathematics

$\int \frac{2 x^2 \cos \left(x^2\right)-\sin \left(x^2\right)}{x^2} d x=$

  • A
    $\frac{\sin \left(x^2\right)}{x^2}+c$
  • B
    $\frac{\cos \left(x^2\right)}{x^2}+c$
  • C
    $\sin \left(x^2\right)+\mathrm{c}$
  • Correct answer
    D
    $\frac{\sin \left(x^2\right)}{x}+c$

Solution

$\begin{aligned} & \text { Let } h=\int \frac{2 x^2 \cos \left(x^2\right)-\sin \left(x^2\right)}{x^2} d x \\ = & \int \frac{x \cdot d\left(\sin \left(x^2\right)\right)-\sin \left(x^2\right) \cdot d(x)}{x^2} \\ = & \int d\left(\frac{\sin \left(x^2\right)}{x}\right)=\frac{\sin \left(x^2\right)}{x}+c .\end{aligned}$

Hence, the correct answer is option: 'D'.

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