Calculus 2: Trigonometric substitution

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.

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.