Problem 20
Question
Find the indefinite integral and check your result by differentiation. $$ \int v^{-1 / 2} d v $$
Step-by-Step Solution
Verified Answer
The indefinite integral of \(v^{-1/2}\) is \(2\sqrt{v} + C\).
1Step 1: Apply the Power Rule for Integration
When the power of the variable v is a real number, the power rule for integration can be applied. This integral should be computed as follows: \[\int v^{-1/2} dv = 2\sqrt{v} + C\] where C is the arbitrary constant of integration.
2Step 2: Differentiate the integral
To verify the computed integral, differentiate it which should result in the initial function. The derivative of \(2\sqrt{v}\) using the power rule for differentiation is \(v^{-1/2}\) . This checks out since it is the same as the integrand function.
Other exercises in this chapter
Problem 20
Use the Log Rule to find the indefinite integral. $$ \int \frac{x^{2}}{3-x^{3}} d x $$
View solution Problem 20
Find the indefinite integral and check the result by differentiation. $$ \int \frac{6 x}{\left(1+x^{2}\right)^{3}} d x $$
View solution Problem 21
Sketch the region bounded by the graphs of the functions and find the area of the region. $$ \begin{aligned} &y=x e^{-x^{2}}, y=0, x=0, x=1\\\ &\begin{gathered}
View solution Problem 21
Use the Log Rule to find the indefinite integral. $$ \int \frac{x^{2}}{x^{3}+1} d x $$
View solution