Integrate $$ \int\mathrm x{e}^{x^2},\mathrm{d}x $$

Substitution: $$u=x^2$$ -> $$ \frac{du}{dx}=2x$$ -> $$\frac{du}{2}=xdx$$ So $$ \int\mathrm x{e}^{x^2},\mathrm{d}x = \frac{1}{2} \int\mathrm {e}^{u},\mathrm{d}u = \frac{1}{2}{e}^{u}+C = \frac{1}{2}{e}^{x^2}+C $$

Could you explain Newton's three laws of motion without using equations, in layman's terms and in 15 minutes?

The Newton's laws of motion are three physical laws (that is, three "rules" that everything in the Universe follows) that explain the motion of things and define the concept of "force". Newton's first law: "Every body persists in its state of being at rest or of moving uniformly straight forward, except insofar as it is compelled to change its state by force impressed." That is: if a still object never interacts with anything, that object will never start moving. If an object in motion never interacts with anything, it will move in a straight line and with the exact same velocity forever and ever. If an object changes its velocity, or doesn't move in a straight line, or suddenly stops or starts moving, it is necessarily interacting with something; that is, a "force" is acting upon the object. Newton's second law: "The alteration of motion is ever proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed." That is: changes in "momentum" (which is simply the mass of an object times its velocity, and what Newton called "motion") are proportional to and happen in the direction of the net force (the sum of all forces) that acts upon the object. We can rewrite Newton's second law as follows: the net force acting upon an object in an instant is equal to the variation of the momentum in that instant (technically speaking, to the time derivative of the momentum). This means that if the mass of the object doesn't change, the net force that acts upon an object is simply its mass times its acceleration. Newton's third law: "To every action there is always opposed an equal reaction: or the mutual actions of two bodies upon each other are always equal, and directed to contrary parts." This one law may look particularly obscure or difficult to understand, but it is not. It simply states that all forces are equal "interactions" between objects. Every force is accompanied by a simultaneous force of equal magnitude and opposite direction, because both objects are part of the same interaction.

What is the difference between a vector and a matrix?

None! Vectors are simply matrices with just one row or one column. You can apply everything you know about matrices to vectors! Be careful, though, not to mistake the functions you use with vectors with those you use with matrices. For example, the dot product is not the same as the matrix product: it is a "special" type of matrix product in which it doesn't matter if the involved vectors are columns or rows: it will always be equivalent to multiplying a row vector with a column vector. Let's see an example. $$\vec{u_{1}}$$ and $$\vec{u_{2}}$$ are two vectors: $$\vec{u_{1}}=\begin{pmatrix}a\\b\end{pmatrix}$$ and $$\vec{u_{2}}=\begin{pmatrix}c\\d\end{pmatrix}$$ The matrix product $$\vec{u_{1}}\vec{u_{2}}=\begin{pmatrix}a\\b\end{pmatrix}\cdot\begin{pmatrix}c\\d\end{pmatrix}$$ does not exist ("it is not defined"), because the number of columns of the first matrix (1) does not match the number of rows of the second matrix (2). However, the dot product $$\vec{u_{1}}\cdot\vec{u_{2}}$$ is $$\vec{u_{1}}\cdot\vec{u_{2}}=\begin{pmatrix}a & b\\\end{pmatrix}\cdot\begin{pmatrix}c\\d\end{pmatrix}=ac+bd$$, which is exactly the matrix product of a one row matrix with a one column matrix. (Note, however, that if $$\vec{u_{2}}=\begin{pmatrix}c & d\\\end{pmatrix}$$, then the matrix product $$\vec{u_{1}}\vec{u_{2}}=\begin{pmatrix}a\\b\end{pmatrix}\cdot\begin{pmatrix}c & d\\\end{pmatrix}=\begin{pmatrix}ac & ad\\bc & bd\end{pmatrix}$$; the matrix product of two vectors can exist and be different from the dot product!).