Solve the integral: ∫ (3)/((x+1)^2 + 4)?

Question

Any ideas to solve this animal? Thanks!

*Obviously the power only applies to (x+1)

Answer 1

Solve the integral: ∫ 3 / [(x+1)² + 4] dx.

Make a substitution. Let
x + 1 = 2tanθ
dx = 2sec²θ dθ
(x + 1)² = 4tan²θ

∫ {3 / [(x+1)² + 4]} dx

= ∫ {3 / (4tan²θ + 4)} (2sec²θ dθ)

= ∫ { 6sec²θ / (4tan²θ + 4)} dθ

= ∫ (6/4) dθ = (3/2)θ + C

x + 1 = 2tanθ
tanθ = (x + 1)/2
θ = arctan[(x + 1)/2]

= (3/2)θ + C = (3/2)arctan[(x + 1)/2] + C

Answer 2

Hey there!

Here's the answer.

∫ 3/(x+1)^2+4 dx --> Write the problem.
3∫ 1/(x+1)^2+4 dx --> Use the property ∫ cf(x) dx=c∫ f(x).
u=x+1, then du=1 --> Use substitution, let x+1 equal u, then the derivative of u is the same as the derivative of x+1, which is 1.
3∫ 1/u^2+4 du --> Substitute u for x+1.
3∫ 1/u^2+2^2 du --> Rewrite 4 as 2^2.
3(1/2arctan(u/2))+C --> Use the property ∫ 1/(u^2+a^2) du=1/a*arctan(u/a)+C.
3(1/2arctan((x+1)/2))+C --> Substitute x+1 for u.
3/2arctan((x+1)/2)+C Distribute 3 into the above expression.

So the answer is 3/2arctan((x+1)/2)+C.

Hope it helps!

Answer 3

A hint: It's an arctangent integral.
To see it, just let u = x+1, du = dx, dx = du
So we get
∫ (3) du / u^2 + 4) =3/2 arctan (u/2) = 3/2 arctan( (x+1)/2 ) + C.

Answer 4

EDIT _________ i do no longer think of there is any much less stressful thank you to combine this. Gohpihan's answer above is the main finished answer. and particular the only thank you to combine it. ______________________________________... enable : S(n) = ? sec? x dx. Then, from the source referred to below, S(n) = [a million/(n-a million)]. sec??² x. tan x + [(n-2)/(n-a million)]. S(n-2) .......... (a million) ______________________________________ for this reason, the mandatory essential is = ? sec³ x tan? x dx dx = ? sec³ x ( tan² x )² dx = ? sec³ x ( sec² x - a million )² dx = ? sec³ x ( sec? x - 2 sec² x + a million ) dx = ? sec? x dx - 2· ? sec? x dx + ? sec³ x dx = S(7) - 2· S(5) + S(3) ...................................... (2) _____________________________ Now, the first component to the respond is carried out. you in basic terms could use (a million) very patiently in (2). that provides you the great very final answer. _____________________________ satisfied to help ! _____________________________

Answer 5

not really any idea. i don't even know what the first symbol is and all i can do is the multiplying and adding hahahah. ∫ 3/(x^2+2x+5)