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Write an expression for the function,f(x), with the properties. f '(x)= \frac{\cos(x)}{x } and...

Question:

Write an expression for the function,f(x), with the properties. {eq}f '(x)= \frac{\cos(x)}{x } {/eq} and f(5)= 5.

{eq}f(x) = \int _{t= a}^{t= b} {/eq} where a =_____ and b = _____

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus Part II (FTC II) states that {eq}\displaystyle \int_a^x f(t)\ dt=F(t)\bigg\vert_a^x =F(x)-F(a), {/eq}

where {eq}\displaystyle F(x) \text{ is the antiderivative of } f(x): F'(x)=f(x) {/eq} and {eq}\displaystyle '=\frac{d}{dx}. {/eq}

Answer and Explanation:

Knowing that {eq}\displaystyle f'(x)=\frac{\cos x}{x} \text{ and } f(5)=5, {/eq} implies that {eq}\displaystyle f(x) {/eq} is the antiderivative of {eq}\displaystyle \frac{\cos x}{x} {/eq}

therefore {eq}\displaystyle \int_a^x \frac{\cos t}{t}\ dt=f(t)\bigg\vert_a^x =f(x)-f(a), \text{ for any constant }a \implies \displaystyle \text{ for } a=5: f(x)=5+\int_5^x \frac{\cos t}{t}\ dt\implies \boxed{a=5, b=x}. {/eq}


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The Fundamental Theorem of Calculus

from Math 104: Calculus

Chapter 12 / Lesson 10
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