Calculus 2

Complete the square for the expression $x^2 + 2x$ and then perform a trigonometric substitution.

Find the indefinite integral of $\displaystyle x \arccos(x) \, dx$ using trigonometric substitution.

Integrate $\displaystyle \int_{0}^{1} \frac{1}{x^2 + 1} \, dx$ using trigonometric substitution.

Evaluate $\displaystyle \int \frac{1}{x^2 + 4} \, dx$ using trigonometric substitution.

Evaluate $\displaystyle \int \frac{1}{9 - x^2} \, dx$ using appropriate trigonometric substitution.

Show that the area of a circle is $\pi \times r^2$ using trigonometric substitution.

Evaluate the integral $\displaystyle \int \left( e^x \, \cdot \, \sqrt{e^{2x} - 4}\right) \, dx$ using substitution.

Evaluate $\displaystyle \int \frac{x}{\sqrt{(x+5)^2 + 4}} \, dx$ using trigonometric substitution after completing the square.

Evaluate $\displaystyle \int \frac{1}{16 - (x-4)^2} \, dx$ using trigonometric substitution after completing the square.

Evaluate the integral $\displaystyle \int \frac{\sqrt{9 - x^2}}{x^2} \, dx$ using trig substitution, where you substitute $x = 3\sin\theta$.

Find the square root of one minus $x^2$.

Find the integral of $\frac{\sqrt{1-x^2}}{x^4} \, dx$ by making the substitution $x = \cos(\theta)$.

Determine the location of the center of mass of a rod with density $\frac{1}{(x^2 + 9)^{3/2}}$ where $x$ goes from 0 to 4.

Find the integral $\displaystyle \int \frac{1}{y^3 \sqrt{y^2 - 1}} \, dy$.

Find the integral $\displaystyle \int \frac{\sqrt{t^2 + 2t}}{t + 1} \, dt$.

Find the integral of cosine of $z$ over the square root of $3 + \cos^2(z) \, dz$.

Integrate $\displaystyle \frac{dx}{(x^2 + 2^2)^{3/2}}$ using the tangent substitution where $x = 2 \tan(\theta)$.

Compute the integral of $\displaystyle \int \sqrt{9 - x^2} \, dx$ using the sine substitution where $x = 3\sin(\theta)$.

Integrate $\frac{dx}{\sqrt{4x^2 - 1}}$ using the secant substitution where $x = \frac{1}{2}\sec(\theta)$.

Integrate $\frac{4}{x^2 - 2x - 8}$ using partial fractions.