Question

integration of (2x(1-x^-3))dx​

Answer 1

ANSWER:

x² + (2/x) + C

GIVEN:

$2x(1 - x^{-3})$

TO INTEGRATE:

$2x(1 - x^{-3})$

FORMUALE:

$\displaystyle \int( {x}^{n} )dx = \frac{ {x}^{n + 1} }{n + 1} + C$

EXPLANATION:

$\displaystyle \int\bigg(2x(1 - x^{ - 3})\bigg)dx \\ \\ \\ \displaystyle 2\int \bigg(x(1 - \frac{1}{ {x}^{3} } )\bigg)dx \\ \\ \\ 2 \int\bigg((x - \frac{x}{ {x}^{3} } ) \bigg)dx \\ \\ \\ 2 \int xdx - 2 \int \frac{1}{ {x}^{2} } dx \\ \\ \\ 2 \bigg( \frac{ {x}^{1 + 1} }{1 + 1} \bigg) + C_1 - 2 \bigg(\frac{ {x}^{ - 2 + 1} }{ - 2 + 1} \bigg) - C_2 \\ \\ \\ 2\bigg(\frac{ {x}^{2} }{2} \bigg) + C_1 - 2\bigg( \frac{ {x}^{ - 1} }{ - 1} \bigg) - C_2 \\ \\ \\ {x}^{2} + C_1 + \frac{2}{x} - C_2 \\ \\ \\ {x}^{2} + \frac{2}{x} + C$

Here C is the contant which is equal to $C_1 - C_2$

$\\ \displaystyle \sf \bf \int2x(1 - x^{ - 3})dx = {x}^{2} + \frac{2}{x} + C$

Answer 2

GIVEN :–

• A function $\: \: { \bold{ 2x(1 - {x}^{ - 3} )}} \: \:$

TO FIND :–

• Integration = ?

SOLUTION :–

• Let the function –

$\\ \implies{ \bold{ I = \int 2x(1 - {x}^{ - 3} ).dx}}$

$\\ \implies{ \bold{ I = \int \{2x -(2x ) {x}^{ - 3} \}.dx}}$

$\\ \implies{ \bold{ I = 2 \int (x - {x}^{ - 2}).dx}}$

• Using identity –

$\\ \: \: \blacktriangleright \: \: { \bold{ \int {x}^{n} .dx = \dfrac{ {x}^{n + 1} }{n + 1} }}$

• So that –

$\\ \implies{ \bold{ I = 2 \left( \dfrac{{x}^{1 + 1} }{1 + 1} - \dfrac{{x}^{ - 2 + 1}}{ - 2 + 1} \right)+c}}$

$\\ \implies{ \bold{ I = 2 \left( \dfrac{{x}^{2} }{2} - \dfrac{{x}^{ - 1}}{ - 1} \right) + c}}$

$\\ \implies \large{ \boxed{ \bold{ I = \left( {{x}^{2} } + \dfrac{2}{ x} \right) + c}}}$

$\\ \rule{220}{2}$

OTHER IDENTITY :–

$\\ \: \: \blacktriangleright \: \: { \bold{ \int [K.F(x)].dx = K \int F(x).dx }}$