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Chapter 5: Problem 100

Simplify the given expressions. $$\frac{d}{d x} \int_{0}^{x^{2}} \frac{d t}{t^{2}+4}$$

Short Answer

Expert verified
Answer: The derivative of the expression is \(\frac{2x}{x^2 + 4}\).

Step by step solution

01

Apply Fundamental Theorem of Calculus (Part 1)

According to the Fundamental Theorem of Calculus (Part 1), we have: $$\frac{d}{d x} \int_{0}^{x^{2}} \frac{d t}{t^{2}+4} = \frac{1}{x^2 + 4} \frac{d(x^2)}{dx}$$
02

Differentiate the argument \(x^2\) with respect to \(x\)

We compute the first derivative of \(x^2\) with respect to \(x\): $$\frac{d(x^2)}{dx} = 2x$$
03

Multiply the derivative of the argument with the value

Now, we substitute the expression we found in Step 2 into the expression from Step 1: $$\frac{1}{x^2 + 4} \frac{d(x^2)}{dx} = \frac{1}{x^2 + 4} (2x)$$
04

Simplify the expression

Multiply the two expressions: $$\frac{1}{x^2 + 4} (2x) = \frac{2x}{x^2 + 4}$$ So, the simplified expression is: $$\frac{d}{d x} \int_{0}^{x^{2}} \frac{d t}{t^{2}+4} = \frac{2x}{x^2 + 4}$$

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Most popular questions from this chapter

Consider the function \(f(x)=x^{2}-4 x\) a. Graph \(f\) on the interval \(x \geq 0\) b. For what value of \(b>0\) is \(\int_{0}^{b} f(x) d x=0 ?\) c. In general, for the function \(f(x)=x^{2}-a x,\) where \(a>0,\) for what value of \(b>0\) (as a function of \(a\) ) is \(\int_{0}^{b} f(x) d x=0 ?\)

Consider the function \(f\) and the points \(a, b,\) and \(c\) a. Find the area function \(A(x)=\int_{a}^{x} f(t) d t\) using the Fundamental Theorem. b. Graph \(f\) and \(A\) c. Evaluate \(A(b)\) and \(A(c)\) and interpret the results using the graphs of part \((b)\) $$f(x)=\cos \pi x ; a=0, b=\frac{1}{2}, c=1$$

Find the area of the following regions. The region bounded by the graph of \(f(\theta)=\cos \theta \sin \theta\) and the \(\theta\) -axis between \(\theta=0\) and \(\theta=\pi / 2\).

Consider the integral \(I(p)=\int_{0}^{1} x^{p} d x\) where \(p\) is a positive integer. a. Write the left Riemann sum for the integral with \(n\) subintervals. b. It is a fact (proved by the 17 th-century mathematicians Fermat and Pascal) that \(\lim _{n \rightarrow \infty} \frac{1}{n} \sum_{k=0}^{n-1}\left(\frac{k}{n}\right)^{p}=\frac{1}{p+1} \cdot\) Use this fact to evaluate \(I(p)\)

Evaluate the following definite integrals using the Fundamental Theorem of Calculus. $$\int_{0}^{\pi / 3} \sec x \tan x d x$$
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