Evaluate these integrals. intsqrt{e^x+1}dx

Yasmin 2020-11-06 Answered
Evaluate these integrals.
ex+1dx
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Expert Answer

Delorenzoz
Answered 2020-11-07 Author has 91 answers

Evaluate:
ex+1dx (1)
Simplification:
Substitute:
ex+1=t
exdx=dt
dx=dtex
dx=dtt1
in(1),
ex+1dx=tt1dt (2)
Now again substitute
t=u
[12tdt=dudt=2tdudt=2u2du]
in (2),
ex+1dx=tt1dt
=2uu21(2u)du
=2u2u21du
=2u21+1u21du
=2(u21u21+1u21)du
=2(1+1u21)du
=2(1+1(u+1)(u1))du (3)
Now evaluate 1(u+1)(u1) using partial fraction method,
1(u+1)(u1)=Au+1+Bu1
1=A(u1)+B(u+1)
Put u=1, 1=2BB=12
Put u=1, 1=2AA=12
1(u+1)(u1)=12(u+1)+12(u1) (4)
Now using (4) in (3),
ex+1dx=2(1+(12(u+1)+12(u1)))du
=2(112(u+1)+12(u1))du
=2uln|u+1|+ln|u1|+C
=2tln|t+1|+ln|t1|+C
=2ex+1ln|ex+1+1|+ln|ex+11|+C
Hence
ex+1dx=2ex+1ln|ex+1+1|+ln|ex+11|+C

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