Question

integral of 1/✓t . cos (✓t + 3) . dt​

Answer 1

$\implies\displaystyle \int \sf{ \frac{1}{ \sqrt{t} } \: cos( \sqrt{t} + 3) } \: dt$

$\implies\displaystyle \int \sf{ \frac{cos( \sqrt{t} + 3) \: dt}{ \sqrt{t} } \: }$

Now we will Integrate the above expression using substitution.

$\sf \: \sqrt{t} + 3 = m$

Now by differentiating we get :-

$\sf \: \dfrac{1}{2} {t}^{ - \frac{1}{2} } dt= dm$

$\sf \: { \dfrac{1}{{ \sqrt{t} }}} \: dt= 2dm$

$\implies\displaystyle \int \sf{ cos( \sqrt{t} + 3)\frac{ \: dt}{ \sqrt{t} } \: }$

$\implies\displaystyle \int \sf{ cos( m) \: 2 \: dm }$

$\implies\displaystyle 2\int \sf{ cos( m) \: dm }$

We know that :-

$\underline{\boxed{\bf{\int \sf{ cos( x) \: dx = \sin(x) + c } }}}$

$\implies\displaystyle 2\int \sf{ cos( m) \: dm } = 2 \sin(m) + c$

now put m = √t +3

$\implies\displaystyle \sf2 \sin( \sqrt{t} + 3 ) + c$

Where c is constant.

Answer :

$\displaystyle \int \sf{ \frac{1}{ \sqrt{t} } \: cos( \sqrt{t} + 3) } \: dt = \displaystyle \sf2 \sin( \sqrt{t} + 3 ) + c$