Calculus II


Indefinite Integrals

Example: Determine the integral

\begin{align} \int te^{\sqrt{t}}\ dt \end{align}

Walpha Query: integral t*e^(sqrt(t))

Shows: The integral (with a Show steps option), plots of the indefinite integral function, alternate forms of the function, and a series expansion at t=0 (with a More terms option).

Other Examples: Not every integral is computed the way they would generally be done by hand. The steps in the following examples are not the usual technique of integration:

\begin{align} \sqrt{1+\left(\frac{2}{3\sqrt[3]{x}}\right)^2} \end{align}
\begin{align} \sin^4(x)\cos^4(x) \end{align}

Definite Integral

Example: Compute the integral

\begin{align} \int_0^1 xe^t\ dt \end{align}

Walpha Query: integral from t=0 to t=1 of x*e^t

Parametric Curves


Example: Plot the parametric curve

\begin{align} x(t)=t^2-1,\quad y(t)=(t-1)^2+2 \end{align}

from t=-2 to t=3.

Walpha Query:

plot [t^2-1,(t-1)^2+2] from t=-2 to t=3

Polar Curves


Example: Plot the polar curve

\begin{align} r(\theta)=5+\sin(8\theta),\quad \theta\in [0,2\pi] \end{align}

Walpha Query: polar plot r=5+sin(8t) from t=0 to t=2*pi

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