Calculus 2

Simplify and integrate $\displaystyle \int \frac{4}{x^2 - 2x - 8} \, dx$ by completing the square.

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

Indefinite integral of $\displaystyle \int \frac{1}{\sqrt{x^2 - 25}} \, dx$ using trigonometric substitution.

Integrate the integral of $\frac{x^3}{\sqrt{1-x^2}}$ with respect to $x$.

Evaluate the integral of $\displaystyle \frac{\sqrt{25x^2 - 4}}{x}$ with respect to $x$.

Evaluate the definite integral $\displaystyle \int_{0}^{3} x \sqrt{x^2 + 9} \, dx$.

Find the area bounded by the ellipse $2x^2 + 9y^2 = 1$.

Solve the integral of $\displaystyle \int \frac{\sqrt{x+2}^3(x^2+4x)}{(x+2)^{-3/2}} \, dx$ by completing the square and using a trigonometric substitution.

Given the integral with a radical, use the substitution $x = 3\sin\theta$ to simplify $\sqrt{9 - x^2}$ and find the integral.

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

Suppose $4x^2 + 1 = \sec^2(\theta)$. Use the substitution $x = \frac{1}{2} \tan(\theta)$ to find the integral.

Given the integral $\displaystyle \int \frac{e^x}{e^{2x} \sqrt{9 - e^{2x}}} \, dx$, use trigonometric substitution to evaluate it.

Evaluate the integral $\displaystyle \int \frac{dx}{(x-3)^2 \sqrt{9 - (x-3)^2}}$.

Make the substitution $u = e^x$ to evaluate the integral $\int \frac{du}{u^2 \sqrt{9 - u^2}}$.

Using the substitution $x = \sqrt{3} \sec(\theta)$, evaluate the integral $\displaystyle \int \frac{\sqrt{x^2 - 3}}{x} \, dx$.

Calculate the definite integral of $\sec^2(\theta) - 1$ from $\theta = 0$ to $\theta = \frac{\pi}{6}$.

Integrate $\cos^3 \theta$ with respect to $\theta$.

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

Find the area of an ellipse using calculus.

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