Question

1/√5-2x + √3-2x,Integrate the given function defined on proper domain w.r.t. x.

Answer 1

Dear Student,

Answer:

$\frac{(3-2x)^{\frac{3}{2} }- (5-2x)^{\frac{3}{2} }}{6} +C$

To do integration of given function

Rationalise the denominator,

$\frac{1}{\sqrt{(5-2x)} +\sqrt{(3-2x)} } .\frac{\sqrt{(5-2x)} -\sqrt{(3-2x)}}{\sqrt{(5-2x)} -\sqrt{(3-2x)}} \\\\\\= \frac{\sqrt{(5-2x)} -\sqrt{(3-2x)}}{5-2x-3+2x} \\ \\ \\ = \frac{\sqrt{(5-2x)} -\sqrt{(3-2x)}}{2}$

∫   $\frac{\sqrt{(5-2x)} -\sqrt{(3-2x)}}{2}$ dx

or

= 1/2 ∫√(5-2x) dx - 1/2 ∫ √(3-2x) dx

put 5-2x = t  => - 2 dx = dt => dx = -1/2 dt

and in second term

put 3-2x = u  => - 2 dx = du => dx = -1/2 du

= -1/4 ∫√t dt + 1/4 ∫√u du

apply power rule of integration

= -1/6(t) ^(3/2) + 1/6(u) ^(3/2) +C

now undo substitution

= -1/6(√(5-2x)) ^(3/2) + 1/6(√(3-2x)) ^(3/2) +C

or

$\frac{(3-2x)^{\frac{3}{2} }- (5-2x)^{\frac{3}{2} }}{6} +C$

is the final answer.

hope it helps you.

Answer 2

In the attachment I have answered this problem.

The integrand is modified in such a way that it is suitable for integration.

See the attachment for detailed solution.