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Chapter 11: Problem 2

Verify the statement by showing that the derivative of the right side is equal to the integrand of the left side. $$ \int \frac{4}{\sqrt{x}} d x=8 \sqrt{x}+C $$

Short Answer

Expert verified
After differentiating the right side of the equation, which is \(8 \sqrt{x}+C\), with respect to x using the power rule, the derived expression is \( \frac{4}{\sqrt{x}}\), matching the original integrand of the left side of the equation. Therefore the verification is successful because the derivative of the right side equals the integrand of the left side.

Step by step solution

01

Identify the Function to Differentiate

The first step is to identify the function that needs to be differentiated. In this case, the right side of the equation, which is \(8 \sqrt{x}+C\), is what needs to be differentiated.
02

Apply the Power Rule for Differentiation

Next step is to differentiate \(8 \sqrt{x}+C\) with respect to x. When you differentiate \(8 \sqrt{x}+C\) with respect to x, use the power rule, which states that if \(f(x) = x^n\), then \(f'(x) = n*x^{n-1}\). Applying the power rule to this function, notice that \(\sqrt{x}\) is the same as \(x^{1/2}\). The derivative of a constant (C) is zero.
03

Compare the Differentiated Function with the Original Integrand

In step 3, the result from the derivative should be compared to the original integrand of the left side of the given expression. The original integrand is \( \frac{4}{\sqrt{x}}\). Re-writing the integral in the form of a power of x makes comparing easier, i.e., \( \frac{4}{x^{1/2}}\).
04

Simplify and Confirm Equality

Simplify the expressions and determine if they are equal. If they are identical, then the verification is successful

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Most popular questions from this chapter

Sketch the region bounded by the graphs of the functions and find the area of the region. $$ y=\frac{1}{x^{2}}, y=0, x=1, x=5 $$

Sketch the region bounded by the graphs of the functions and find the area of the region. $$ f(x)=\frac{1}{x}, g(x)=-e^{x}, x=\frac{1}{2}, x=1 $$

Use the Midpoint Rule with \(n=4\) to approximate the area of the region bounded by the graph of \(f\) and the \(x\) -axis over the interval. Compare your result with the exact area. Sketch the region. $$ f(x)=x(1-x)^{2} $$ $$ [0,1] $$

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Numerical Approximation Use the Midpoint Rule and the Trapezoidal Rule with \(n=4\) to approximate \(\pi\) where \(\pi=\int_{0}^{1} \frac{4}{1+x^{2}} d x\) Then use a graphing utility to evaluate the definite integral. Compare all of your results.
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