Evaluate the integral. \int \sec^{2}x\tan x dx

socorraob 2021-12-19 Answered
Evaluate the integral.
sec2xtanxdx
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Expert Answer

Edward Patten
Answered 2021-12-20 Author has 38 answers

Step 1
Given: I=sec2xtanxdx
for evaluating given integral, we substitute
tanx=t...(1)
now, differentiating equation (1) with respect to x
ddx(tanx)=ddx(t)   (ddx(tanx)=sec2x)
sec2x=dtdx
sec2xdx=dt
Step 2
now, replacing sec2xdx with dt, tanx with t in given integral
so,
sec2xtanxdx=tdt   (xndx=xn+1n+1+c)
=(t22)+c...(2)
now, replacing t with tanx in equation (2)
so,
sec2xtanxdx=tan2x2+c
hence, given integral is equal to tan2x2+c.

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Jack Maxson
Answered 2021-12-21 Author has 25 answers
sec2(x)tan(x)dx
Substitution u=sec2(x)dudx=2sec2(x)tan(x)dx=12sec2(x)tan(x)du:
=121du
Now we calculate:
1du
Integral of a constant:
=u
We substitute the already calculated integrals:
121du
=u2
Reverse replacement u=sec2(x):
=sec2(x)2
sec2(x)tan(x)dx
=sec2(x)2+C
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nick1337
Answered 2021-12-28 Author has 510 answers

sec2(x)tan(x)dx
Apply u-substitution: u=tan(x)
=udu
Apply the Power Rule
=u22
Substitute back u=tan(x)
=tan2(x)2
Add a constant to the solution
=tan2(x)2+C

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