Question

сalculate the indefinite integral


detailed solution

Answer 1

Step-by-step explanation:

take 4 in common then it's Direct formula

Answer 2

Answer:

$\bold\red{\frac{1}{2} {sin}^{ - 1} 2 x + c}$

Step-by-step explanation:

we have to integrate,

$\int \frac{1}{ \sqrt{1 - 4 {x}^{2} } } dx$

Taking 4 common,

we get,

$\int \frac{1}{ \sqrt{4( \frac{1}{4} - {x}^{2}) } } dx \\ \\ = \frac{1}{2} \int \frac{1}{ \sqrt{ \frac{1}{4} - {x}^{2} } } dx \\ \\ = \frac{1}{2} \int \frac{1}{ \sqrt{ { (\frac{1}{2}) }^{2} - ( {x}^{2}) } } dx$

Now,

we know that,

$\int \frac{dx}{ \sqrt{ {a}^{2} - {x}^{2} } } = {sin}^{ - 1} \frac{x}{a} + c$

Therefore,

we get,

Integration

$= \frac{1}{2} {sin}^{ - 1} \frac{x}{ \frac{1}{2} } + c \\ \\ = \bold{\frac{1}{2} {sin}^{ - 1} 2 x + c}$

where,

c is an arbitrary constant