Find the indefinite integral. \int \frac{-1}{\sqrt{1-(4t+1)^{2}}}dt

Deragz 2021-12-17 Answered
Find the indefinite integral.
11(4t+1)2dt
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Expert Answer

Wendy Boykin
Answered 2021-12-18 Author has 35 answers
Step 1
We have the given integral,
11(4t+1)2dt
By substituting u=4t+1, that implies du=4dt
Therefore, the integral becomes,
=11u2(14)du
=1411u2du
Step 2
Now, by substituting sinx=u, that implies, cosxdx=du
Therefore, the integral becomes,
=1411sin2x(cosx)dx
=141cos2x(cosx)dx
=141cosx(cosx)dx
=14dx
=14x+C
Step 3
But we know, sinx=u, therefore x=sin1(u)
By substituting this value of x,
=14sin1u+C
Then by resubstituting u=4t+1,
=14sin1(4t+1)+C

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Tiefdruckot
Answered 2021-12-19 Author has 46 answers
11(4t+1)2dt
141u2du
1411u2du
14arcsin(u)
14arcsin(4t+1)
arcsin(4t+1)4
Add C
Solution:
arcsin(4t+1)4+C

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nick1337
Answered 2021-12-28 Author has 575 answers

Step 1
Given:
11(4t+1)2dt
Substitution u=4t+1dudt=4
=1411u2du
This is the well-known tabular integral:
=arcsin(u)
We substitute the already calculated integrals:
1411u2du
=arcsin(u)4
Reverse replacement u=4t+1:
=arcsin(4t+1)4
Answer:
=arcsin(4t+1)4+C

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