Question

Q] Integrate by using the Substitution in bracket :

$\displaystyle \sf\int \limits \dfrac{ {9r}^{2} }{ \sqrt{1 - {r}^{3} } } \, dr,(use \: u = 1 - {r}^{3)}$
​

Answer 1

Given

$\to \sf \int \dfrac{9 {r}^{2} }{ \sqrt{1 - {r}^{3} } } dr$

Using Substitution Method ,Let

$\to \sf \: 1 - {r}^{3} = u$

Now Differentiate on both side wrt r

$\sf \to \dfrac{d(1 - {r}^{3} )}{dr} = \dfrac{du}{dr}$

$\sf \to \: - 3 {r}^{3 - 1} = \dfrac{du}{dr}$

$\sf \to \: - 3 {r}^{2} = \dfrac{du}{dr}$

$\sf \to \: (3 {r}^{2} )dr = - du$

Now we can write as

$\to \sf3 \int \dfrac{3 {r}^{2} }{ \sqrt{1 - {r}^{3} } } dr$

Put the value

$\sf \to \: 3\int \dfrac{ - du}{ \sqrt{u} }$

$\sf \to \: - 3\int \dfrac{ du}{ \sqrt{u} }$

$\sf \to \: - 3\int \dfrac{ du}{ {u} {}^{ \frac{1}{2} } }$

$\sf \to \: - 3 \int ({u}^{ \frac{ - 1}{2} } )du$

$\sf \to \: - 3 \bigg( \dfrac{u {}^{ \frac{ - 1}{2} + 1 } }{ \dfrac{ - 1}{2} + 1} \bigg) + c$

$\sf \to \: - 3 \bigg( \dfrac{ {u}^{ \frac{ - 1 + 2}{2} } }{ \dfrac{ - 1 + 2}{2} } \bigg) + c$

$\sf \to - 3 \bigg( \dfrac{u {}^{ \frac{1}{2} } }{ \dfrac{1}{2} } \bigg) + c$

$\sf \to - 3(2 {u}^{ \frac{ 1 }{2} } ) + c$

$\sf \to \: - 6( \sqrt{u} ) + c$

Now put the value

$\to \sf \: 1 - {r}^{3} = u$

We get

$\sf \to - 6( \sqrt{1 - {r}^{3} } ) + c$

Answer

$\sf \to - 6( \sqrt{1 - {r}^{3} } ) + c$

Answer 2

$\sf \: given \: \: \boxed{ \sf u = 1 - {r}^{3} } \\ \ \\ \displaystyle \sf\int \limits \dfrac{ {9r}^{2} }{ \sqrt{1 - {r}^{3} } } \, dr \\ \\ \\ \sf \ \frac{d(1 - {r}^{3} )}{dr} = \frac{du}{dr} \\ \\ \\ \sf \: - 3 {r}^{3 - 1} = \frac{du}{dr} \\ \\ \\ \sf \: - 3 {r}^{2} = \frac{du}{dr} \\ \\ \\ \sf \: 3 \int \frac{ {3r}^{2} }{ \sqrt{1 - {r}^{3} } } dr \\ \\ \\ \sf \: - 3 \int{ \frac{ du}{ \sqrt{u} } } \\ \\ \\ \sf \: - 3 \int \: \frac{du}{ {u}^{ \frac{1}{2} } } \\ \\ \ \\ \sf \: - 3 \int( {u}^{ \frac{ - 1}{2}) } du \\ \\ \\ \sf \: - 3( \frac{ {u}^{? \frac{ - 1 + 2}{2} } }{ - \frac{1}{2} + 2} ) + c \\ \\ \\ \sf \: - 3( \frac{ {u}^{ \frac{1}{2} } }{ \frac{1}{2} } ) + c \\ \\ \\ \sf \: - 3(2 {u}^{ \frac{1}{2} } )+ c \\ \\ \\ \sf \: - 6 \sqrt{u} + c \\ \\ \sf \: put \: value \: of \: u : - \\ \\ \sf \: - 6 \sqrt{1 - {r}^{3} } + c$

hope it helps you dear...