Calculus Answers

True or False.

The value of the integral "\\displaystyle{\\int_{1}^{2}x(x^2-1)^8 dx}" is an integer

Which of the following represents the integral  "\\displaystyle{\\int \\frac{x}{\\sqrt{1+x^2}}dx}"

1) "\\dfrac{x^2}{\\sqrt{1+x}}+C"

2) "\\ln{(1+x^2)}+C"

3) "\\sqrt{1+x^2}+C"

4) "\\dfrac{x^2}{\\sqrt{1+x^2}}+C"

Which of the following values represent the definite "{\\int_{\\pi\/2}^{2\\pi\/3}\\frac{1}{1+\\cos{\\theta}}d\\theta}"

1) e

2) 1

3) "\\sqrt{3}"

4) "\\sqrt{3}-1"

What is the u substitution choice for "{\\int \\frac{\\sin{\\sqrt{x}}}{\\sqrt{x}}dx}"

1) u=x

​

2) "u=\\dfrac{1}{\\sqrt{x}}"

3) u=x

4) None of the above

Which of the following values represent the definite integral

"\\intop\\begin{matrix}\n \\pi\/4 \\\\\n 0 \n\\end{matrix}" "e" ^tanθ sec² θ dθ

1) e²

2) e - 1

3) e/2

4) e

Which of the following values represent the definite integral 

"\\intop\\begin{matrix}\n 1 \\\\\n 0 \n\\end{matrix}" "sqrt(1- x^2) dx"

1) "\\pi\/2"

2) "\\pi\/3"

3) "\\pi\/3"

4) "3\\pi\/4"

2. Prove that ∇ · (A ~ + B~ ) = ∇A~ + ∇B~

Find an interval where the following functions has a root. Show all you work how you chose the interval. i) f(x) = 3x + sin(x) − e^x

The area bounded by the curves y=x²+2 and y=3x+2 from intersection point to intersection point is rotated around x=3. Determine the formula for the solid of revolution which is obtianed from this rotation. Explain your logic.

Consider the function 𝑓(𝑥, 𝑦) = 𝑥𝑦 ( 𝑥 2−𝑦 2 𝑥 2+𝑦2 ) , (𝑥, 𝑦) ≠ (0, 0). Show that lim (𝑥,𝑦)→(0,0) 𝑓(𝑥, 𝑦) = 0