Calculus 2

Find the anti-derivative of $\cos^3(x)$.

Find the anti-derivative of $\cos^5(x)$.

Find the indefinite integral of $\displaystyle \int \cos^5(x) \cdot \sin(x) \, dx$.

Find the indefinite integral of $\sin^5(x) \cdot \cos^2(x) \, dx$.

Find the indefinite integral of $\sin^5(x) \cdot \cos^3(x)$.

Find the indefinite integral of $extstyle \\sin^2(x)$.

Find the anti-derivative of $\cos^2(3x)$.

Find the indefinite integral of $\\displaystyle \\int \\sin^4(x) \, dx$.

What is the anti-derivative of $\sin^2(x)\cos^2(x)\, dx$?

Find the indefinite integral of $\cos^2(x) \cdot \tan^3(x)\, dx$.

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 $x^3 \sqrt{1 - 4x^2}$.

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.

Evaluate the definite integral $\displaystyle \int_{4}^{6} \frac{x^2}{\sqrt{x^2 - 9}} \, dx$ by using trigonometric substitution in two different ways: first by computing the antiderivative and then using the Fundamental Theorem of Calculus with the original limits of integration; second by changing the limits of integration after substituting $x = 3 \sec(\theta)$ and evaluating the integral in terms of $\theta$.

Solve $\displaystyle \int \frac{\sec \theta \cdot 4 \tan^2 \theta}{\cos^2 \theta} \, d\theta$ using a substitution method.

Find the length of the curve $y=\frac{1}{6}x^3+\frac{1}{2x}$ from $x=1$ to $x=5$.