True or False. If F(x) = \int_{-2}^{3x} \sin(t)\ dt, then the second fundamental theorem of...

Question:

True or False.

If {eq}F(x) = \int_{-2}^{3x} \sin(t)\ dt {/eq}, then the second fundamental theorem of calculus can be used to evaluate F '(x) as follows {eq}F '(x) = \sin (3x). {/eq}

Second Fundamental Theorem of Calculus:

The second fundamental theorem of calculus states that the differential of {eq}F(x) = \displaystyle \int_{a}^{x} f(t) \, \mathrm{d}t {/eq} is {eq}F'(x) = f(x) {/eq}.

Suppose the upper limit of integration is a general function of {eq}x {/eq} denoted by {eq}u(x) {/eq}.

Then, the derivative uses the chain rule and is given by

{eq}\displaystyle F'(x) = f(x) \frac{\mathrm{d}u}{\mathrm{d}x} {/eq}

Answer and Explanation:

Note that the derivative of {eq}\displaystyle F(x) = \int_{-2}^{3x} \sin(t)\ \mathrm{d}t {/eq} is simply {eq}F'(x) = \sin(x) {/eq} if the upper limit of integration is {eq}x {/eq}.

But, the upper limit of integration is {eq}3x {/eq} so we will use the chain rule, along with the second fundamental theorem of calculus, to differentiate it:

{eq}\begin{align*} \displaystyle F(x) &= \int_{-2}^{3x} \sin(t)\ \mathrm{d}t\\ \displaystyle F'(x) &=\sin(3x) \frac{\mathrm{d}}{\mathrm{d}x}(3x)\\ \displaystyle F'(x) &=\sin(3x)(3)\\ \displaystyle F'(x) &=3\sin(3x)\\ \end{align*} {/eq}

Thus, the given statement is false as the derivative of {eq}\displaystyle F(x) = \int_{-2}^{3x} \sin(t)\ \mathrm{d}t {/eq} is {eq}3\sin(3x) {/eq} and not {eq}\sin(3x) {/eq}.


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The Fundamental Theorem of Calculus

from Math 104: Calculus

Chapter 12 / Lesson 10
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