Calculus 2

Set up a right triangle based on the expression $4-x^2$ to use trigonometric substitution for integration, identifying which side represents the hypotenuse.

Simplify the integral using trigonometric substitution and express the result back in terms of $x$.

Evaluate the integral $\displaystyle \int x^3 \sqrt{9 - x^2} \, dx$ using trigonometric substitution.

Evaluate the integral $8 \displaystyle \int_{1}^{\frac{\sqrt{5}}{2}} u^2 \, du$.

Evaluate the integral $\displaystyle \int_{1}^{2} \frac{\sqrt{x^2 - 1}}{x} \, dx$.

Evaluate the integral $\displaystyle \int_{0}^{\pi/2} \frac{\cos t}{\sqrt{1 + \sin^2 t}} \, dt$.

$\displaystyle \int \frac{\sqrt{x^2-25}}{x} \, dx$ 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.

Find the anti-derivative of $\frac{x - 9}{x^2 + 3x - 10}$ using partial fractions.

Evaluate the integral from 0 to 2 of $\displaystyle \int_{0}^{2} (5x^4 - 2x) \, dx$ using the fundamental theorem of calculus Part 2.

Find the limit as $x \to 0$ of $\displaystyle \frac{\sin x - x}{x^2}$ using L'Hôpital's Rule.

Evaluate the integral of $\displaystyle \frac{1}{x^2 + 1}$ using partial fractions.

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 $\displaystyle \int \cos^2(\theta) \, d\theta$ using the double angle identity.

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

Evaluate the integral of $\frac{1}{\sqrt{1-x^2}}$ which is equivalent to the inverse sine of x.

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