Calculus 2: Trigonometric substitution

$\displaystyle \int \frac{\sqrt{x^2-25}}{x} \, dx$ with $x = 5 \sec \theta$

$\displaystyle \int \sqrt{3 - x^2} \, dx$

Evaluate the definite integral from $4$ to $4\sqrt{3}$ of $\frac{1}{x^2 \sqrt{x^2 + 16}} \, dx$.

Integrate $\frac{1}{\tan^2 \theta} \cdot \sec \theta \cdot \sec^2 \theta \, d\theta$ by rewriting the expression in terms of sines and cosines and using a trigonometric substitution.

Solve the integral of $\frac{dx}{\sqrt{x^2 - 9}}$.

Evaluate $\displaystyle \int \sec^3 \theta \, d\theta$ using trig substitution where $x = \tan(\theta)$.

Evaluate the integral of the square root of $1-x^2$ using trigonometric substitution.

For the integral $\displaystyle \sqrt{a^2-x^2}$, make the trigonometric substitution $x=a\sin\theta$ and find the differential $dx=a\cos\theta\,d\theta$.

Evaluate the integral of the form $\displaystyle \int \frac{dx}{\sqrt{x^2 - a^2}}$ by making the substitution $x = a \sec\theta$.

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 sine substitution where the substitution is $x = 6 \sin(\theta)$.

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

Using tangent substitution, where $5x=2\tan(\theta)$, solve an integral with a coefficient on $x^2$.

Using secant substitution, where $x = 4 \sec(\theta)$, solve an integral with a rational power, such as a fractional power of three halves.

Evaluate the integral using trigonometric substitution: $\displaystyle \int \sqrt{1-x^2} \, dx$.

Using a secant substitution, simplify and integrate $\frac{1}{x \sqrt{x^2 - 4}}$.

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

Simplify the integral $\displaystyle \, \int \frac{1}{\sqrt{16 - x^2}} \, dx$ using the substitution $x = 4 \sin \theta$.

Simplify the integral $\displaystyle \int \frac{1}{\sqrt{9 - x^2}} \, dx$ using the substitution $x = 3 \sin \theta$.