Use this only Power Formula Logarithm Formula Exponential Functions Trigonometric

Maaghu

Maaghu

Answered question

2021-11-28

Use this only Power Formula Logarithm Formula Exponential Functions Trigonometric Functions. Complete solution pls
012x3+x2+3(x2+1)2dx

Answer & Explanation

Lounctirough

Lounctirough

Beginner2021-11-29Added 14 answers

Given 012x3+x2+3(x2+1)2dx
Perf orm partial decomposition:
=2x+1x2+1+22x(x2+1)2dx
Apply lienarity:
=2x+1x2+1dx2x1(x2+1)2dx
Now solve
2x+1x2+1dx
Expand:
=2xx2+1dx+1x2+1dx
Substitute u=x2+1dudx=2xdx=12xdu
=121udu
We knov that 1xdx=ln(x)
=ln(u)2
Undo the substitution:
=ln(x2+1)2
We know that 1x2+1dx=arctan(x)
Plug in solved integrals:
=ln(x2+1)+arctan(x)
Now solvind: x1(x2+1)2
Expand:
=x(x2+1)2dx1(x2+1)2dx
Substitute u=x2+1dudx=2xdx=12xdu

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