Question

Let $f\left(x\right)=\int 2xcos\left(x^2\right)dx$. Find the value of $f\left(\sqrt{\pi }\right)$, if f(0) = 0

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Answer 1

We want to find the integral of the function $2xcos\left(x^2\right)$. We know the integral of the function cosx, but not that of $cos\left(x^2\right)$ or $2xcos\left(x^2\right)$ We can see that the derivative of $x^2$, i.e 2x, is also part of the given function. This is the hint for making the substitution $x^2=t$. Differentiating this, we get ,
$dt=2xdx\Rightarrow \int 2xcos\left(x^2\right)dx=\int cos\left(t\right)dt=sint+C$.
We will now replace t with $x^2$.
So we get, $f\left(x\right)=sin\left(x^2\right)+C$
We are given,
$f\left(0\right)=0\Rightarrow C=0$
We want to find $f\left(\sqrt{\pi }\right)$
$f\left(\sqrt{\pi }\right)=sin{\left(\sqrt{\pi }\right)}^2=sin\pi =0$