Question

x√x+3,Integrate the given function defined on proper domain w.r.t. x.

Answer 1

HELLO DEAR,




YOUR QUESTIONS IS--------------∫x/√(x + 3).dx


put (x + 3) = t Also, [x = t - 3]
=> dx = dt


therefore, ∫(t - 3)/√t.dt

=> ∫(√t - 3/√t).dt

=> 3/2t^{3/2} - 3(2)t^{1/2} + c.

=> 3/2t^{3/2} - 6t^½ + c.


I HOPE ITS HELP YOU DEAR,
THANKS

Answer 2

Dear student,

Solution:

∫ x √(x+3) dx

let x+3 = t

dx = dt

so, x = t -3

apply substitution

∫( t -3) √t dt

or ∫( t √t - 3√t ) dt

apply linearity

∫ $t^{\frac{3}{2} } dt$ - 3 ∫√t dt

Apply power rule for integration

$= \frac{2 t^{\frac{5}{2} } }{5} -3 \frac{2 t^{\frac{3}{2} } }{3} +c$

undo substitution:

$= \frac{2(x+3)^{\frac{5}{2} } }{5} -2(x+3)^{\frac{3}{2} } +c$

Hope it helps you.