Calculus 2

Convert $x = \sec\theta$ back in terms of $x$ using a right triangle and basic SOHCAHTOA.

For a radical $\sqrt{x^2 - 1}$, use trigonometric substitution and translate $\sin\theta$ back to $x$ in the problem solved.

Integrate using trig substitution when you have both a radical expression in the numerator and a coefficient on the $x^2$ term.

Solve the integral using trigonometric substitution where the square root involves $1 - \sin^2(x)$.

Evaluate the integral using trigonometric substitution.

Perform a trigonometric substitution for evaluating the integral involving inverse substitution where $x=\frac{1}{2}u$.

Evaluate the integral using trigonometric substitution where $x = 3 \sin \theta$ for the expression involving $\sqrt{9 - x^2}$.

Using trigonometric substitution, solve integrals that have integrals involving $a^2 - u^2$, $a^2 + u^2$, and $u^2 - a^2$ inside the radical.

Using trigonometric substitution, solve integrals involving $a^2 + u^2$ under the radical.

Using trigonometric substitution, simplify the expression $\sqrt{a^2 - u^2}$, where $u = a \sin\theta$.

Solve the indefinite integral $\displaystyle \int 4x^2 \sqrt{16 - x^2} \, dx$ using appropriate substitution.

Using trigonometric identities, such as $\cos^2(\theta) + \sin^2(\theta) = 1$, find related identities to simplify expressions in integral problems.

Identify the type of trigonometric substitution needed (sine, tangent, secant) to integrate a function involving expressions such as $a^2 - x^2$, $a^2 + x^2$, or $x^2 - a^2$, complete the necessary substitutions, and integrate the resulting expression.

Find the length of the curve from one point to another using integration and calculus techniques for calculating arc length.

Using integration, find the exact length around the curve from point A to point B for a given function f(x).

Evaluate the integral $\int_{0}^{\infty} f(x) \, dx$ by breaking it into $\int_{0}^{1} f(x) \, dx$ and $\int_{1}^{\infty} f(x) \, dx$, addressing the infinite interval and discontinuity at zero.

Calculate the volume of a solid of revolution by using the disc and shell methods for a given region in a plane spun about an axis.

Integrate a rational function using partial fractions.

Solve a first order linear ordinary differential equation using the integrating factor method.

Solve the differential equation using the method of integrating factors, where the initially given differential equation is linear with the coefficient functions for $y$ and $y'$ dependent on $x$.