Integrate the functions.

1. \(\displaystyle\int 4 \; dx\)

2. \(\displaystyle\int du\)

Find \(\displaystyle\int 6x^2 \; dx\)

Integrate

1. \(\displaystyle\int\left(4x^2+8x-7\right)dx\)

2. \(\displaystyle\int\left(5x^\frac{2}{3}-4\right)dx\)

Find \(\displaystyle\int\left(3e^x-5x\right)dx\)

1. Find \(\displaystyle\int\dfrac{7}{x}\;dx\)

2. Suppose it is known that \(x>0\). Find \(\displaystyle\int \left(6e^x+\frac{4}{x}-1\right)dx\)

3. Suppose it is known \(x>0\). Find \(\displaystyle\int\dfrac{x^4+3x^3+5}{x}\)

Find \(f(x)\) if \(f'(x)=3x^2-4x+2\) and \(f(1)=4\).

Evaluate \(\displaystyle\int\dfrac{12x+4}{6x^2+4x+1}dx\)

Evaluate \(\displaystyle\int\dfrac{x}{3x^2+5}dx\)

Evaluate \(\displaystyle\int3e^{3x+7}\,dx\)

Evaluate \(\displaystyle\int4xe^{5x^2-2}\,dx\)

Evaluate \(\displaystyle\int10(9x+1)(9x^2+2x+7)^8\,dx\)

Evaluate \(\displaystyle\int\dfrac{x^3}{\sqrt[3]{8x^4-5}}\,dx\)

Evaluate \(\displaystyle\int x\sqrt{x+3}\; dx\)

1. \(\displaystyle\int e^{6x}\; dx\)

2. \(\displaystyle\int e^{\frac{2}{3}x}\; dx\)

Evaluate \(\displaystyle\int_3^5 (x^2+3x-2)\;dx\)

Evaluate \(\displaystyle\int_4^{10}\dfrac{1}{x-3}\, dx\)

Evaluate \(\displaystyle\int_4^{10}\dfrac{1}{x-3}\, dx\)

Use four rectangles to approximate the area of the planar region under the graph of \[f(x)=0.3x^3-1.8x^2+15\] over the interval \([1,6]\). (See Figure 5)

Find the area bounded by the function \[f(x)=0.3x^3-1.8x^2+15\] the \(x\)-axis, and the lines \(x=1\) and \(x=6\). (This is our function from Example 1)

Find the value of \(\displaystyle\int^5_2(x^2-5x+4)\, dx\)

Find the area of the region bounded by \(f(x)=x^2-5x+4\), the \(x\)-axis, and the lines \(x=2\) and \(x=5\).

Evaluate \(\displaystyle\int^\infty_5\frac{1}{x^\frac{3}{2}}\, dx\)

Evaluate \(\displaystyle\int^1_{-\infty}\frac{1}{(x-2)^3}\, dx\)

Evaluate \(\displaystyle\int^\infty_{-\infty}x\, dx\)

Determine whether or not the integral \(\displaystyle\int^1_{-1}\frac{1}{x^2}\, dx\) converges.

Suppose a pollutant is seeping into the ground near a dump site at the rate of \[f(t)=\frac{5\text{,}400}{(t+1.5)^3}\] liters per month, where \(t\) denotes the time from now in months. If the seepage of the pollutant continues indefinitely into the future, what is the total amount of pollutant that will seep into the ground?

Suppose the concentration of a particular drug that is administered orally is given by \(C_{po}(t)=100\left(e^{-0.5t}-e^{-0.8t}\right)\). The concentration of that drug when administered intravenously is given by \(C_{iv}150e^{-0.5t}\). This function considers both absorption and elimination of the drug over time. Determine the bioavailability, \(F\), of the drug.

Evaluate \(\displaystyle\int x^4\ln{x}\, dx\)

Evaluate \(\displaystyle\int xe^{4x}\, dx\)

Evaluate \(\displaystyle\int \ln{x}\, dx\)

A manufacturer estimates that the relationship between the number of months, \(t\), from now and the number, \(N\), (in thousands) of units of a product she can produce per month is given by \[N(t)=14te^{-0.08t}\] Find the function that estimates the total production by the manufacturer if the total production initially is \(0\).