Question

i need full explanation

Answer 1

Given: I =  $\int\ {\frac{(x+3)^{3} }{3} } \, dx$

To FInd: value of I

Solution: $I=$ $\int\ {\frac{(x+3)^{3} }{3} } \, dx$

    Let   $x+3 = y$   --------(1)

      on differentiating equation (1) wrt x

              $1+0=\frac{dy}{dx}$⇒  $dy=dx$ ------------(2)

on putting value of y from equation (1) in $I$

                $I=$   $\int\ {\frac{y^{3} }{3} } \, dy$

using formula, $\int\ {x^{n} } \, dx = \frac{x^{n+1} }{n+1}$

             ⇒$I=\frac{1}{3} \int\ {y^{3} } \, dy$  

             ⇒ $I=$    $\frac{1}{3} \frac{y^{3+1} }{(3+1)}$

        ⇒ $I=\frac{y^{4} }{12}$ $= \frac{(x+3)^{4} }{12}$