Integrate term by term using the power rule. The integral of x^3 dx = x^4/4, the integral of 2x dx = x^2, and the integral of 1 dx = x. Adding a single constant of integration, the integral of (x^3 + 2x + 1) dx = x^4/4 + x^2 + x + C.

integral of x^n dx = x^(n+1)/(n+1) + C

The standard integrals are: the integral of sin x dx = -cos x + C; the integral of cos x dx = sin x + C; and the integral of e^x dx = e^x + C. Also, the integral of sec^2 x dx = tan x + C. These are obtained by reversing the corresponding derivative formulas.

integral of sin x dx = -cos x + C

(1) The integral from a to b of f(x) dx = - the integral from b to a of f(x) dx (interchanging the limits changes the sign). (2) The integral from a to b of f(x) dx = the integral from a to c of f(x) dx + the integral from c to b of f(x) dx (splitting the interval at a point c). (3) The integral from 0 to a of f(x) dx = the integral from 0 to a of f(a - x) dx. Also, for an odd function the integral from -a to a is 0, and for an even function it is twice the integral from 0 to a.

integral(a to b) f(x) dx = -integral(b to a) f(x) dx

Integrate term by term. The integral of (1/x) dx = ln|x| and the integral of e^x dx = e^x. Adding the constant of integration, the integral of (1/x + e^x) dx = ln|x| + e^x + C.

integral of (1/x) dx = ln|x| + C

The area bounded by the curve y = f(x), the x-axis, and the ordinates x = a and x = b (with f(x) >= 0) is given by the definite integral: Area = the integral from a to b of y dx = the integral from a to b of f(x) dx. The area between two curves y = f(x) (upper) and y = g(x) (lower) from x = a to x = b is Area = the integral from a to b of [f(x) - g(x)] dx.

Area = integral(a to b) f(x) dx

If a region is symmetric about the x-axis, the y-axis or the origin, we can compute the area of one symmetric part and then multiply by the appropriate factor to get the total area. For example, a circle x^2 + y^2 = a^2 is symmetric about both axes, so its total area equals 4 times the area of the part lying in the first quadrant. Using symmetry reduces the amount of integration needed and simplifies the calculation.

A differential equation is an equation that involves an independent variable, a dependent variable and the derivatives of the dependent variable with respect to the independent variable. The order of a differential equation is the order of the highest derivative appearing in it. The degree of a differential equation is the power (exponent) of the highest order derivative, provided the equation is a polynomial in the derivatives (free from radicals and fractions in the derivatives).

The highest order derivative present in the equation is d2y/dx2 (the second derivative), so the order of the differential equation is 2. The power (exponent) of this highest order derivative is 3, and the equation is a polynomial in the derivatives, so the degree is 3. Hence the order is 2 and the degree is 3.

The general solution of a differential equation is the solution that contains as many arbitrary constants as the order of the differential equation; it represents a whole family of curves. A particular solution is obtained from the general solution by giving definite values to these arbitrary constants using given initial or boundary conditions; it represents one specific member of the family.

A differential equation of the form dy/dx = f(x, y) is called homogeneous if the function f(x, y) can be written as a function of the ratio y/x only (that is, f depends on x and y only through y/x). To solve it, we use the substitution y = v x (so v = y/x), which gives dy/dx = v + x (dv/dx). This substitution reduces the equation to a variable separable form in v and x, which can then be integrated.

y = v x, dy/dx = v + x(dv/dx)

A scalar quantity is one that has only magnitude and no direction (for example, mass, length, temperature). A vector quantity is one that has both magnitude and direction and obeys the laws of vector addition (for example, displacement, velocity, force). The position vector of a point P with coordinates (x, y, z) with respect to the origin O is the vector OP = x i + y j + z k, and its magnitude is sqrt(x^2 + y^2 + z^2).

|OP| = sqrt(x^2 + y^2 + z^2)

The magnitude of a = 2 i + 3 j + 6 k is |a| = sqrt(2^2 + 3^2 + 6^2) = sqrt(4 + 9 + 36) = sqrt(49) = 7. A unit vector in the direction of a is obtained by dividing a by its magnitude: a hat = a/|a| = (1/7)(2 i + 3 j + 6 k) = (2/7) i + (3/7) j + (6/7) k.

a hat = a/|a|

A unit vector is a vector whose magnitude is 1; it is used to specify direction. A zero (null) vector is a vector whose magnitude is zero and whose direction is indeterminate; it is denoted by 0. Collinear vectors are two or more vectors that are parallel to the same line, irrespective of their magnitudes and directions; such vectors can be written as scalar multiples of one another.

The scalar projection (component) of vector a on vector b is given by (a . b)/|b|. It represents the length of the shadow of a along the direction of b. If the dot product a . b = 0 (and neither vector is zero), then cos theta = 0, so theta = 90 degrees; this means the two vectors are perpendicular (orthogonal) to each other.

projection of a on b = (a . b)/|b|

Two vectors are perpendicular if their dot product is zero. So a . b = (2)(1) + (lambda)(-2) + (1)(3) = 2 - 2 lambda + 3 = 5 - 2 lambda. Setting this equal to zero: 5 - 2 lambda = 0, so 2 lambda = 5 and lambda = 5/2.

a . b = 0 for perpendicular vectors

The direction cosines of a line are the cosines of the angles that the line makes with the positive directions of the x-, y- and z-axes; they are usually denoted by l, m and n. The direction ratios are any set of numbers proportional to the direction cosines. The direction cosines always satisfy the relation l^2 + m^2 + n^2 = 1.

l^2 + m^2 + n^2 = 1

The direction cosines are obtained by dividing the direction ratios by the square root of the sum of their squares. Here the magnitude = sqrt(1^2 + 2^2 + 2^2) = sqrt(1 + 4 + 4) = sqrt(9) = 3. So the direction cosines are l = 1/3, m = 2/3 and n = 2/3. We can check l^2 + m^2 + n^2 = 1/9 + 4/9 + 4/9 = 9/9 = 1.

l = a/sqrt(a^2+b^2+c^2), etc.

The general Cartesian equation of a plane is Ax + By + Cz + D = 0, where A, B, C are the direction ratios of the normal to the plane and are not all zero. The equation of a plane that makes intercepts a, b and c on the x-, y- and z-axes respectively is x/a + y/b + z/c = 1 (the intercept form). A plane is uniquely determined by three non-collinear points.

Ax + By + Cz + D = 0

The angle between two planes is equal to the angle between their normals. If the planes are A1 x + B1 y + C1 z + D1 = 0 and A2 x + B2 y + C2 z + D2 = 0, then cos theta = |A1 A2 + B1 B2 + C1 C2| / (sqrt(A1^2 + B1^2 + C1^2) x sqrt(A2^2 + B2^2 + C2^2)). The two planes are perpendicular if A1 A2 + B1 B2 + C1 C2 = 0, and parallel if A1/A2 = B1/B2 = C1/C2.

cos theta = |A1A2 + B1B2 + C1C2|/(|n1||n2|)

For two parallel planes Ax + By + Cz + D1 = 0 and Ax + By + Cz + D2 = 0, the distance between them is |D1 - D2|/sqrt(A^2 + B^2 + C^2). Here A = 2, B = -2, C = 1, D1 = 3, D2 = 9. Distance = |3 - 9|/sqrt(2^2 + (-2)^2 + 1^2) = 6/sqrt(4 + 4 + 1) = 6/sqrt(9) = 6/3 = 2 units.

d = |D1 - D2|/sqrt(A^2 + B^2 + C^2)