`f(t) = sin(t) + 2sinh(t)` Find the most general antiderivative of the function. (Check your answer by differentiation.)

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Chapter 4, 4.9 - Problem 18 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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sbernabel | Student, Graduate | (Level 1) Adjunct Educator

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`int f(t) dt=int sin(t)+2 sinh(t) dt = int sin(t) dt + 2 int sinh(t) dt `

We note that the derivative of `cosh(t)=sinh(t) =>d/dx cosh(t) = sinh(t)=>d/dx sinh(t)=cosh(t)`

This means that

`int sinh(t) dt = cosh(t) +c_2 ` . We already know the anti-derivative of `sin(t) ` , therefore the integral of `f(t) ` is as follows:

`int f(t) dt = -cos(t) + c_1 + 2 cosh(t) + c_2 = -cos(t) + cosh(t) + c ` , where `c=c_1+c_2 ` is a constant. To check that this solution works we differentiate,

`d/dx (-cos(t)+cosh(t)+c)=-(-sin(t))+sinh(t)+0=sin(t)+sinh(t) ` as we intended.

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