Calculus Answers

Consider the integral "\\displaystyle{\\int \\frac{2}{x^2-2x}dx}=\\displaystyle{\\ln{\\frac{x-a}{x}}+C}"

 Which of the following values represent the constant a ?

1. 1
2. 7
3. 4
4. 2

True or False

To evaluate the integral "\\displaystyle{\\int \\sqrt{x}\\ln{(x)}dx}"  by integration by parts,  it is wise to choose "u=\\sqrt{x}" and "dv=\\ln{(x)}dx"

True or False:

 Consider the integral "\\displaystyle{\\int \\frac{x^3+x}{x-1}}dx" To evaluate the integral, we need to first perform the long division.

True or False:

The integral "\\displaystyle{\\int x\\cos{(\\pi x)} dx}" cannot be evaluated using the integration by parts

Evaluate the integral "\\displaystyle{\\int_{0}^{1}te^{t}dt}"

1, 2 ,3 or 4?

Evaluate the integral "\\displaystyle{\\int_{0}^{1}\\frac{x}{x+1}dx}"

1. "1-\\ln{ 2}"
2. "\\ln{2}"
3. 0
4. "-\\ln{2}"

If f(0) = g(0) = 0 and f'' and g'' are continuous, show that

the integral of f(x)g''(x)dx (from 0 to a) = f(a)g'(a) - f'(a)g(a) + the integral of f''(x)g(x)dx (from 0 to a).

state and prove the necessary conditions for the existence of maximum value and minimum value?