Calculus 1

Use U-Substitution to evaluate the following integral

$\displaystyle\int 4x(x^2 + 5)^{50} \ dx$

Evaluate the following integral

$\displaystyle\int\frac{\sin (\ln (x))}{x} \ dx$

Evaluate the following integral

$\displaystyle\int x^3 \sin (x^4 + 2) \ dx$

Evaluate the indefinite integral

$\displaystyle\int (x^4 + 2)^4 4x^3 \ dx$

Evaluate $\displaystyle\int (\ln{x})^4 \frac{2}{x} \ dx$

Find the area under the curve over the interval $[1,4]$

$f(x) = \frac{2}{x}$

Prove the fundamental theorem of calculus

Compute the definite integrals

$\displaystyle\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \tan (x) \ dx$ and $\displaystyle\int_{\frac{-\pi}{3}}^{\frac{\pi}{3}} \tan (x) \ dx$

Find the area bounded by the following curves/lines

$y = x + 1$

$y = 9 - x^2$

$x = -1$

$x = 2$

Find the area between the curves $y = x$ and $y = x^2$

Compute the area of the region bounded by the curves $y = x^3$ and $y = 3x - 2$

Find the average value on $[0, 16]$ of $f(x) = \sqrt{x}$

What is the average value of the function $f(x) = 3x^2 - 2x$ on $[1, 4]$

Use the shell method to determine the volume formed by the bounded region rotated about the x-axis.

$y = x^2$, $y = 0$, $x = 2$