Question

Evaluate the indefinite integral using substitution method.

Answer 1

$\frac{2(3 - x)^{ \frac{5}{2} } }{5} - \frac{2(3 - x)^{ \frac{3}{2} } }{3} + C$is the answer

Step-by-step explanation:-

Given:

${\int {(x-2)\sqrt{3-x}} \ dx }$

Differentiate on both sides,

${\int {(x-2)\sqrt{3-x}} \ dx },u = 3 - x$

Isolate and substitute back,

${\int {(x-2)\sqrt{3-x}} \ \times ( - 1) du },u = 3 - x$

Substitute back,

${\int ( - ( - u + 3 - 2) \sqrt{u})du }$

Applying property of integral ∫ kf(x)dx = k ∫ f(x)dxdx,

$- ∫ ( - u + 3 - 2) \sqrt{u} \: du,u = 3 - x$

Combining like terms,

$- ∫ ( - u +1) \sqrt{u} \: du,u = 3 - x$

Now applying distributive property,

$- ∫ (( - u +1) \sqrt{u}) \: du,u = 3 - x$

$= > - ∫ ( - \sqrt{u} \times u + \sqrt{u} )\: du,u = 3 - x$

Converting to exponential form,

$- ∫ ( - u^{ \frac{1}{2} } \times u + u^{ \frac{1}{2} } )\: du,u = 3 - x$

Multiplying the first two monomuals after integral, we get,

$- ∫ ( - u^{ \frac{3}{2} } + u^{ \frac{1}{2} } )\: du,u = 3 - x$

Now applying the prperty of ∫ f(x) + g(x)dx = ∫ f(x)dx + ∫ g(x)dx,

$- ( - ∫u^{ \frac{3}{2} } \: du + ∫u^{ \frac{1}{2} } \: du),u = 3 - x$

Now integrate the power rule,

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

Divide the fractions by multiplying its reciprocals,

$- ( - u^{ \frac{5}{2} } \times \frac{2}{5} + u^{ \frac{3}{2} } \times \frac{2}{3}),u = 3 - x$

Now write as single fractions,

$- ( - \frac{2u^{ \frac{5}{2} } }{5} + \frac{2u^{ \frac{3}{2} } }{3})$

Removing the parentheses,

$\frac{2u^{ \frac{5}{2} } }{5} - \frac{2u^{ \frac{3}{2} } }{3},u = 3 - x$

Substitute back,

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

Now adding the constant of integration C∈R,

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

Hence, the answer.