Question

Evaluate
∫ sin√x/√x dx​

Answer 1

Solution:-

$\to \rm \: \int \dfrac{ \sin\sqrt{x} }{ \sqrt{x} } dx$

Now we can write as

$\rm \to \int \sin\sqrt{x} \times \dfrac{1}{ \sqrt{x} } \: dx$

Using substitution method

$\rm let \: \sqrt{x} = t$

$\rm \: ( \sqrt{x} ) \dfrac{dx}{dt} = tdt$

$\rm \: \dfrac{1}{2 \sqrt{x} } dx = dt$

Now take

$\rm \to \int \sin\sqrt{x} \times \dfrac{1}{ \sqrt{x} } \: dx$

Now multiply and divide by 2

$\rm \to2 \int \sin\sqrt{x} \times \dfrac{1}{ 2\sqrt{x} } \: dx$

So we have

$\rm \: \dfrac{1}{2 \sqrt{x} } dx = dt$

We get

$\rm \: 2\int \sin t \: dt$

so integration of of sinx is - cos x

$\rm \: 2( - cos \: t) + c$

$\rm \: - 2 \cos \: t + c$

Now put the value of t

$\rm \: - 2 \cos \sqrt{x} + c$

Answer:-

$\rm \: - 2 \cos \sqrt{x} + c$