Problem 72
True or False? Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. $$ \int_{-1}^{2} \frac{1}{x} d x=[\ln |x|]_{-1}^{2}=\ln 2-\ln 1=\ln 2 $$
Problem 72
Find the indefinite integral using the formulas of Theorem 4.24 \(\int \frac{d x}{(x+2) \sqrt{x^{2}+4 x+8}}\)
Problem 72
(a) integrate to find \(F\) as a function of \(x\) and (b) demonstrate the Second Fundamental Theorem of Calculus by differentiating the result in part (a). $$ F(x)=\int_{1}^{x} \frac{1}{t} d t $$
Problem 72
Let \(s(x)\) and \(c(x)\) be two functions satisfying \(s^{\prime}(x)=c(x)\) and \(c^{\prime}(x)=-s(x)\) for all \(x .\) If \(s(0)=0\) and \(c(0)=1,\) prove that \([s(x)]^{2}+[c(x)]^{2}=1\)
Problem 72
Evaluate the definite integral. Use a graphing utility to verify your result. $$ \int_{1}^{2} e^{1-x} d x $$
Problem 73
Find the indefinite integral using the formulas of Theorem 4.24 \(\int \frac{1}{1-4 x-2 x^{2}} d x\)
Problem 73
Graph the function \(f(x)=\frac{x}{1+x^{2}}\) on the interval \([0, \infty)\) (a) Find the area bounded by the graph of \(f\) and the line \(y=\frac{1}{2} x\) (b) Determine the values of the slope \(m\) such that the line \(y=m x\) and the graph of \(f\) enclose a finite region. (c) Calculate the area of this region as a function of \(m\).
Problem 73
Verify the natural log rule \(\int \frac{1}{x} d x=\ln |C x|\) \(C \neq 0,\) by showing that the derivative of \(\ln |C x|\) is \(1 / x\)
Problem 73
Evaluate the definite integral. Use a graphing utility to verify your result. $$ \int_{1}^{3} \frac{e^{3 / x}}{x^{2}} d x $$
Problem 73
In Exercises \(73-78,\) use the Second Fundamental Theorem of Calculus to find \(F^{\prime}(x)\). $$ F(x)=\int_{-2}^{x}\left(t^{2}-2 t\right) d t $$
