Chapter 4: Q. 41 (page 399)

Use the Second Fundamental Theorem of Calculus, if needed, to calculate each of the derivatives given below.
$\frac{d}{d x} \int_{1}^{8} \ln t d t$

Short Answer

Expert verified

Ans: $\frac{d}{d x} \int_{1}^{8} \ln t d t = 0$

Step by step solution

Step 1. Given information.

given,

$\frac{d}{d x} \int_{1}^{8} \ln t d t$

Step 2. The objective is to calculate the derivative.

Now, if $f$ is continuous on $[a, b]$ then for all $x \in [a, b]$ ,

$\frac{d}{d x} \int_{a}^{u (x)} f (t) d t = f (u (x)) u^{'} (x)$

So,

role="math" $\begin{aligned} f (u (t)) = \ln t \\ f (u (8)) = \ln 8 \\ u^{'} (x) = 0 \\ f (u (x)) u^{'} (x) = \ln 8 (0) = 0 \end{aligned}$

Step 3. The derivative expression can be written as,

$\begin{aligned} \frac{d}{d x} \int_{1}^{8} \ln t d t \\ = 0 [f (u (x)) u^{'} (x) = 0] \end{aligned}$

Therefore, the value is $0$

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Use the Second Fundamental Theorem of Calculus, if needed, to calculate each of the derivatives given below.
$\frac{d}{d x} \int_{1}^{8} \ln t d t$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. The objective is to calculate the derivative.

Step 3. The derivative expression can be written as,

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Use the Second Fundamental Theorem of Calculus, if needed, to calculate each of the derivatives given below. ddx∫18 ln⁡tdt

Short Answer

Step by step solution

Step 1. Given information.

Step 2. The objective is to calculate the derivative.

Step 3. The derivative expression can be written as,

One App. One Place for Learning.

Most popular questions from this chapter

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Use the Second Fundamental Theorem of Calculus, if needed, to calculate each of the derivatives given below.
$\frac{d}{d x} \int_{1}^{8} \ln t d t$