Calculus/Integration

1.

The process of indefinite integration finds a function (F) whose derivative is a given function (f), call it f(V). If

$$\frac{dp(V)}{dV} = f(V)$$

then

$$p(V) = \int f(V)dV + C$$

where C is a constant of integration.

If $f(V) = aV^2 + b \ell n V$ then

$$p(V) = \int f(V)dV = a \frac{V^3}{3} + b(V \ell n V - V) + C$$

(since $\frac{d\left ( a \frac{V^3}{3} + b(V \ell n V - V) + C\right)}{dV} = f(V)$) where, again, C is a constant of integration (which vanishes upon differentiation of p(V) to re-obtain f(V).

2.

The process of definite integration means, given a function f(V) and two values of V, finding the area under the graph of f(V) between the aforementioned values of V:

$$\int_{V=a}^{V=b} f(V)dV = \left . p(V) \right \vert _a^b = p(b) - p(a)$$

where, of course, the constant of integration cancels.

3.

a special integral is defined:

$$\int_1^y \frac{dz}{z} \equiv \ell n y ; y > 0$$ 0 \end{displaymath}">

4.

There exist a special set of indefinite and definite integrals which should be known:

(a)

$$\int_0^\infty e^{-\alpha^2 x} dx = \frac{1}{\alpha^2}$$

(b)

$$\int_0^\infty e^{-\alpha^2 x^2} dx = \frac{1}{2} \sqrt{\frac{\pi}{\alpha^2}}$$

(c)

$$\int \sin ^2 x dx = \int \left ( \frac{e^{\imath x} - e^{-\imath x}}{2 \imath} \right )^2 dx$$

The process of indefinite integration finds a function (F) whose derivative is a given function (f), call it f(V). If

$$\frac{dp(V)}{dV} = f(V)$$

then

$$p(V) = \int f(V)dV + C$$

where C is a constant of integration.

If $f(V) = aV^2 + b \ell n V$ then

$$p(V) = \int f(V)dV = a \frac{V^3}{3} + b(V \ell n V - V) + C$$

(since $\frac{d\left ( a \frac{V^3}{3} + b(V \ell n V - V) + C\right)}{dV} = f(V)$) where, again, C is a constant of integration (which vanishes upon differentiation of p(V) to re-obtain f(V). The process of definite integration means, given a function f(V) and two values of V, finding the area under the graph of f(V) between the aforementioned values of V:

$$\int_{V=a}^{V=b} f(V)dV = \left . p(V) \right \vert _a^b = p(b) - p(a)$$

where, of course, the constant of integration cancels. a special integral is defined:

$$\int_1^y \frac{dz}{z} \equiv \ell n y ; y > 0$$ 0 \end{displaymath}">

There exist a special set of indefinite and definite integrals which should be known:

1.

$$\int_0^\infty e^{-\alpha^2 x} dx = \frac{1}{\alpha^2}$$

2.

$$\int_0^\infty e^{-\alpha^2 x^2} dx = \frac{1}{2} \sqrt{\frac{\pi}{\alpha^2}}$$

3.

$$\int \sin ^2 x dx = \int \left ( \frac{e^{\imath x} - e^{-\imath x}}{2 \imath} \right )^2 dx$$