Integral sinh(x)

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Proof: Integral sinh(x)

(Math | Calculus | Integrals | Table Of | sinh x)

Discussion of (integral)

sinh x dx = cosh x + C.

1. Proof

Strategy

sinh x =

e^x - e^-x2

sinh x dx = (integral)

e^x - ex2

set
u = - x
then we find
du = - dx

substitute du= - dx, u= - x

= (1/2) e^x + (1/2) (integral) - ex dx
= (1/2) e^x + (1/2) e^u du

solve the right integral
= (1/2) e^x + (1/2) e^u + C

substitute back u= - x
= (1/2) e^x + (1/2) ex + C

= e^x + ex2 + C

which by definition
= cosh x + C
Q.E.D.