Q114 / 160 · AP EAMCET 2021 (25 Aug Shift 2) · Mathematics

$\int \cos \sqrt{x} d x$ is equal to

  • Correct answer
    A
    $2 \sqrt{x} \sin \sqrt{x}+2 \cos \sqrt{x}+c$
  • B
    $2 \sqrt{x} \sin \sqrt{x}+2 \sin \sqrt{x}+c$
  • C
    $2 \sqrt{x} \sin \sqrt{x}-2 \cos \sqrt{x}+c$
  • D
    $\sqrt{x} \cos \sqrt{x}-2 \sin \sqrt{x}+c$

Solution

$\int \cos \sqrt{x} d x$ $\begin{aligned} & \text { Let } I=\int \cos \sqrt{x} d x \\ & \text { Let } \sqrt{x}=t \\ & x=t^2\end{aligned}$ Differentiating w.r.t ‘x’ $\begin{aligned} 1 & =2 t \frac{d t}{d x} \\ d x & =2 t d t \\ I & =\int \cos t \cdot 2 t d t \\ I & =2 \int t \cos t d t\end{aligned}$ On applying by-part method $\begin{aligned} & I=2\left[t \int \cos t d t-\int\left\{\frac{d}{d t} t \int \cos t d t\right\} d t\right] \\ & I=2[t \sin t+\cos t]+c \\ & I=2 \sqrt{x} \sin \sqrt{x}+2 \cos \sqrt{x}+c\end{aligned}$

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

Relevant Questions