91Ó°ÊÓ

Evaluate the integrals. $$ \int_{1}^{2} x(x-2)^{5} d x $$

Find \(f(x)\) if \(f(0)=1\) and the tangent line at \((x, f(x))\) has slope \(x\). HINT [See Example 5.]

Decide on what substitution to use, and then evaluate the given integral using a substitution. HINT [See Example 1.] $$ \int \frac{3 e^{-1 / x}}{x^{2}} d x $$

Decide on what substitution to use, and then evaluate the given integral using a substitution. HINT [See Example 1.] $$ \int \frac{2 e^{2 / x}}{x^{2}} d x $$

Find \(f(x)\) if \(f(1)=1\) and the tangent line at \((x, f(x))\) has slope \(\frac{1}{x}\). HINT [See Example 5.]

The rate of U.S. per capita sales of bottled water for the period 2000-2008 can be approximated by \(s(t)=0.04 t^{2}+1.5 t+17\) gallons per year \(\quad(0 \leq t \leq 8)\) where \(t\) is the time in years since the start of \(2000 .{ }^{20}\) Use a Riemann sum with \(n=5\) to estimate the total U.S. per capita sales of bottled water from the start of 2003 to the start of 2008\. (Round your answer to the nearest gallon.) HINT [See Example 3.]

Evaluate the integrals. $$ \int_{1}^{2} x(x-2)^{1 / 3} d x $$

Find \(f(x)\) if \(f(0)=0\) and the tangent line at \((x, f(x))\) has slope \(e^{x}-1\).

Decide on what substitution to use, and then evaluate the given integral using a substitution. HINT [See Example 1.] $$ \int \frac{e^{x}+e^{-x}}{2} d x \text { HINT [See Example 4(b).] } $$

Evaluate the integrals. $$ \int_{0}^{1} x \sqrt{2 x+1} d x $$