`f''(x) = x^3 + sinh(x), f(0) = 1, f(2) = 2.6` Find `f`.

Textbook Question

Chapter 4, 4.9 - Problem 44 - Calculus: Early Transcendentals (7th Edition, James Stewart).
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Borys Shumyatskiy | College Teacher | (Level 3) Associate Educator

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Integrate this once (all integrals are the table ones):

`f'(x)=(1/4)x^4+cosh(x)+C_1`

and twice:

`f(x)=(1/20)x^5+sinh(x)+C_1*x+C_2.`

Now determine the constants `C_1` and `C_2` :

`f(0)=C_2=1.`

`f(2.6)=2^5/20+sinh(2)+C_1*2+1 approx 6.2+2*C_1 = 2.6,` i.e.

`C_1 approx 1.3-3.1=-1.8.`

The answer: approximately `f(x)=x^5/20+sinh(x)-1.8x+1.`

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