Calculus 2: 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 $\displaystyle \int_{0}^{3} x \sqrt{x^2 + 9} \, dx$.

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.

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

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

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

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

Find the area of an ellipse using calculus.

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

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

Evaluate the integral $\displaystyle \int_0^{\frac{3\sqrt{3}}{2}} \frac{x^2}{(4x^2 + 9)^{3/2}} \, dx$ using trigonometric substitution and solve it as a definite integral.

Using trigonometric substitution, solve the integral $\displaystyle\int \frac{1}{\sqrt{36-x^2}} \, dx$.

Evaluate the integral of $\frac{1}{\sqrt{x^2 + 6x}}$ from 1 to 2.

Evaluate the definite integral $\displaystyle \int_0^{\frac{3}{2}} \frac{1}{\sqrt{9-x^2}} \, dx$ using trigonometric substitution.

Evaluate the integral $\displaystyle \int \frac{dx}{\sqrt{x^2 + 2x + 145}}$.

Using trigonometric substitution, find the integral of $\frac{1}{\sqrt{x^2 + 4}}$ with respect to $x$.

Evaluate the indefinite integral of $x^3 \sqrt{1 - 4x^2}$.