Evaluate the indefinite integral. ∫sec2(x)dxx

Harold Kessler

Harold Kessler

Answered

2021-12-30

Evaluate the indefinite integral.
sec2(x)dxx

Answer & Explanation

vicki331g8

vicki331g8

Expert

2021-12-31Added 37 answers

Step 1
We have to evaluate the integral:
sec2(x)dxx
We will use substitution method since derivatives of one function is present in the integral.
So assuming,
t=x
differentiating with respect to x,
dtdx=dxdx
dtdx=12x
2dt=dxx
Substituting above values, we get
sec2(x)dxx=sec2(t)(2dt)
=2sec2(t)dt
Step 2
Since we know that
sec2xdx=tanx+C
therefore,
2sec2(t)dt=2tan(t)+C
=2tan(x)+C
Where, C is an arbitrary constant.
Hence, value of given integral is 2tan(x)+C.
usumbiix

usumbiix

Expert

2022-01-01Added 33 answers

sec2(x)xdx
Substitution u=xdudx=12xdx=2xdu
=2sec2(u)du
sec2(u)du
This is the well-known tabular integral:
=tan(u)
We substitute the already calculated integrals:
2sec2(u)du
=2tan(u)
Reverse replacement u=x:
=2tan(x)
Solution:
=2tan(x)+C
karton

karton

Expert

2022-01-04Added 439 answers

sec(x)2xdx
2sec(t)2dt
2×sec(t)2dt
2tan(t)
2tan(x)
Add CR
Solution:
2tan(x)+C,CR

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