Question

Step by step integration of 1/(5-4x-x^2)^(3/2)

Answer 1

looks the thing inside is a perfect square

Answer 2

start by writing it as square and see if u can cancel out radical

Answer 3

It doesn't work, you need to do trig substitution but I can't figure out the details of how that works yet.

Answer 4

ahh ok its not a perfect square

Answer 5

but still completing the square is the first thing to do here

Answer 6

after completing the square, u wil see wat u need to substitute

Answer 7

\(\large \int \frac{1}{(5-4x-x^2)^{\frac{3}{2}}} dx\)

Answer 8

I have integral (1/(9-(x+2)^2)^(3/2))

Answer 9

And then you use sin for trig substitution but that's where I got stuck

Answer 10

\(\large \int \frac{1}{(5-4x-x^2)^{\frac{3}{2}}} dx\)

\(\large \int \frac{1}{(9 - (x+2)^2)^{\frac{3}{2}}} dx\)

Answer 11

substitute

\(x+2 = 3\sin \theta\)

Answer 12

Could you explain why you use that?

Answer 13

good question :)
plugging that gives me 9sin^2 term,

Answer 14

9 - 9sin^2

Answer 15

i can pull out 9, and simplify the remaining this to cos^2

Answer 16

and cancel out the radical !

Answer 17

plugin first, then u wil understand

Answer 18

Oh okay, thanks a lot! :)

Answer 19

u wil get use to these after doing 2-3 problems

Answer 20

if u dont want to guess like this, there is another way

Answer 21

let me show u that as well

Answer 22

\(\large \int \frac{1}{(5-4x-x^2)^{\frac{3}{2}}} dx\)

\(\large \int \frac{1}{(9 - (x+2)^2)^{\frac{3}{2}}} dx\)

Answer 23

factor out 9 from the bottom

Answer 24

\(\large \int \frac{1}{(5-4x-x^2)^{\frac{3}{2}}} dx\)

\(\large \int \frac{1}{(9 - (x+2)^2)^{\frac{3}{2}}} dx\)

\(\large \int \frac{1}{[9 (1 - \frac{(x+2)^2}{9}]^{\frac{3}{2}}} dx\)

Answer 25

\(\large \int \frac{1}{(5-4x-x^2)^{\frac{3}{2}}} dx\)

\(\large \int \frac{1}{(9 - (x+2)^2)^{\frac{3}{2}}} dx\)

\(\large \int \frac{1}{[9 (1 - \frac{(x+2)^2}{9}]^{\frac{3}{2}}} dx\)

\(\large \int \frac{1}{[9 (1 - (\frac{x+2}{3})^2]^{\frac{3}{2}}} dx\)

Answer 26

Now you're forced to substitute the thing inside square to sin or cos right ?

Answer 27

\(\frac{x+2}{3} = \sin\theta\)

Answer 28

which gives the same substitution :)

Answer 29

Thank you so much :)

Answer 30

do watever way makes more sene to u

Answer 31

np.. u wlc :)