Calculus Solutions, Important Examples, Formulas and Videos

Integration

  • Let  \frac{\mathrm{d} }{\mathrm{d} x}F(x)=f(x) . Then we write  \int f(x)dx=F(x)+c These integrals are called indefinite integrals or general integrals, C is called constant of integration. All these integrals differ by a constant
  • From the geometric point of view , an indefinite integral is collection of family of curves, each of which is obtained by translating one of the curves parallel to itself upwards or downwards along the y -axis.
  • Some properties of indefinite integrals are as follows
  • \int \left [ f(x) +g(x)\right ]=\int f(x)dx+\int g(x)dx
  • For any real number k, \int kf(x)dx=k\int f(x)dx
    More generally if f_{1},f_{2},f_{3}........f_{n} are function and k_{1},k_{2},k_{3}........k_{n} are real numbers . \int \left [ k_{1}f_{1}(x) +k_{2}f_{2}(x)\right ]+.......+\int\left [ k_{n}f_{n}(x) \right ]dx=k_{1}\int f_{1}(x)dx+k_{2}\int f_{2}(x)dx+.....+k_{n}\int f_{n}xdx

Some Standard Integral:

  • \int x^{n}dx=\frac{x^{n+1}}{n+1}+c=,n\neq -1 . Particularly \int dx=x+c
  •  \int \cos xdx=\sin x+c
  •  \int \sin xdx=-\cos x+c
  • \int \sec ^{2}xdx=\tan x+c
  • \int \cos ec ^{2}xdx=-\cot x+c
  • \int \sec x\tan xdx=\sec x+c
  • \int \cos ec x\cot xdx=-\cos ec x+c
  • \int \frac{dx}{\sqrt{1-x^{2}}}=-\sin^{-1}x+c
  • \int \frac{dx}{\sqrt{1-x^{2}}}=-\sin^{-1}x+c
  • \int \frac{dx}{\sqrt{1-x^{2}}}=-\cos^{-1}x+c
  • \int \frac{dx}{{1+x^{2}}}=\tan^{-1}x+c
  • \int \frac{dx}{{1+x^{2}}}=-\cot^{-1}x+c
  •  \int e^{x}dx=e^{x}+c
  • \int a^{x}dx=\frac{a^{x}}{loga}+c
  • \int \frac{dx}{x\sqrt{x^{2}+1}}=\sec ^{-1}x+c
  • \int \frac{dx}{x\sqrt{x^{2}+1}}=-\cos ec ^{-1}x+c
  • \int \frac{1}{x}dx=\log |x|+c
  • Integration by partial fractions:
    Recall that a rational function is ratio of two polynomials of the form , \frac{P(x)}{Q(x)} where P(x) and Q(x) are polynomials in
    X and Q(x) ≠ 0. If degree of the polynomial P(x) is greater than the degree of the polynomial Q(x), then we may divide p(x) by q(x) so

that \frac{P(x)}{Q(x)}=T(x)+\frac{p_{1}(x)}{Q(x)} where T (x) is a polynomial in x and degree of P1(x) is less than the degree of Q(x). T(x) being polynomial can be easily integrated. \frac{p_{1}(x)}{Q(x)} can be integrated by expression \frac{p_{1}(x)}{Q(x)} as the sum of partial fraction of the following type:

  • \frac{px+q}{(x-a)(x-b)}=\frac{A}{(x-a)}+\frac{B}{(x-b)},a\neq b
  • \frac{p+q}{(x-a)^{2}}=\frac{A}{(x-a)}+\frac{b}{(x-a)^{2}}
  • \frac{px^{2}+qx+r}{(x-a)(x-b)(x-c)}=\frac{A}{(x-a)}+\frac{B}{(x-b)}+\frac{C}{(x-c)}
  • \frac{px^{2}+qx+r}{(x-a)^{2}(x-b)}=\frac{A}{(x-a)}+\frac{B}{(x-a)^{2}}+\frac{C}{(x-b)}
  • \frac{px^{2}+qx+r}{(x-a)(x^{2}+bx+c)}=\frac{A}{(x-a)}+\frac{Bx+C}{(x^{2}+bx+c)} ,
    Where , x^{2}+bx+c can not be factorised further.
  • Integration by substitution:
    A change in the variable of integration often reduces an integral to one of the fundamental integrals. The method in which we change the variable to some other variable is called the method of substitution. When the integrand involves some trigonometric functions, we use some well known identities to find the integrals. Using substitution technique, we obtain the following standard integrals.
  • \int \tan xdx=\log |\sec x|+c
  • \int \cot xdx=\log |\sin x|+c
  • \int \sec xdx=\log |\sec x+\tan x|+c
  • \int \cos ec xdx=\log |\cos ec x-\tan x|+c
  • Integrals of some special functions:
    \int \frac{dx}{x^{2}-a^{2}}=\frac{1}{2a}log\left | \frac{x-a}{x+a} \right |+c
    \int \frac{dx}{a^{2}-x^{2}}=\frac{1}{2a}log\left | \frac{a-x}{a+x} \right |+c
    \int \frac{dx}{a^{2}+x^{2}}=\frac{1}{a}\tan^{-1}\frac{x}{a}+c
    f\frac{dx}{\sqrt{x^{2}-a^{2}}}=\log\left |x+\sqrt{x^{2}-a^{2}}\right |+c
    f\frac{dx}{\sqrt{x^{2}-a^{2}}}=\sin^{-1}\frac{x}{a}+c
    f\frac{dx}{\sqrt{x^{2}+a^{2}}}=\log\left |x+\sqrt{x^{2}+a^{2}}\right |+c
  • Integration by parts:
    For given function is f_{1}\: and\: f_{2} we have
    \int f_{1}(x)\cdot f_{2}(x)dx=f_{1}(x)\int f_{2}xdx-\int \left [ \frac{\mathrm{d} }{\mathrm{d} x}f_{1}(x)\cdot \int f_{2}(x)dx \right ]dx , i.e The integral of the product of two functions = first function × integral of the second function – integral of {differential coefficient of the first function × integral of the second function}. Care must be taken in choosing the first function and the second function. Obviously , we must take that function as the second function whose integral is well known to us.
    \int \left [ e^{x}f(x)+f'(x)\right ]=\int e^{x}f(x)dx+c
  • Integrals of the type \int \frac{dx}{ax^{2}+bx+c}\: or\: \int \frac{dx}{\sqrt{ax^{2}+bx+c}} can be transformed in to standard form by expressing
    ax^{2}+bx+c=a\left [ x^{2}+\frac{a}{b}x+\frac{c}{a}\right ]=a\left [ \left ( x+\frac{b}{2a} \right )^{2}+\left ( \frac{c}{a}-\frac{b^{2}}{4a^{2}} \right )\right ]
  • Integral of the type: \int \frac{px+qdx}{ax^{2}+bx+c}\: or\: \int \frac{px+qdx}{\sqrt{ax^{2}+bx+c}} can be transformed in to standard form by expressing
    px+q=A\frac{\mathrm{d} }{\mathrm{d} x}(ax^{2}+bx+c)+B=A(2ax+b)+B
    Where A and B are determined by comparing coefficients on both sides
  • First fundamental theorem of integral calculus: Let the area function be defined by A(x) = \int_{a}^{x}f(x)dx or all x ≥ a, where the function f is assumed to be continuous on [a, b]. Then A′(x) = f(x) for all x∈[a, b]
  • Second fundamental theorem of integral calculus Let f be a continuous function of x defined on the closed interval [a, b] and let F be the function such that \frac{\mathrm{d} }{\mathrm{d} x}F(x)=f(x) for all x in the domain of f, then \int_{a}^{b}f(x)dx=[F(x)+c]_{a}^{b}=F(b)-F(a)
    This is called the definite integral of f over the range [a, b], where a and bare called the limits of integration,
    a being the lower limit and b the upper limit.

 

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