The concept of "capture of integral" is closely accompanied by finding of primitive function. The F (x) function is called an antiderivative to f(x) if its derivative F’(x) equals f (x). As the derivative of any constant is equal to zero, and antiderivatives at f(x) will be infinite much. All of them coincide among themselves to within a constant. Traditional designation of uncertain integral is presented in fig. 1.
It is required to you
- Table of the simplest integrals.
Instruction
1. In mathematics there is quite large number of the ways "take" integral. In this article those from them which it is accepted to call the simplest methods of integration are shortly considered. These receptions use properties of uncertain integrals and identical transformations of subintegral function.
2. 1. Direct integration. Direct integration consists in calculation of integrals by means of their certain properties and special tables. Example 1. To calculate integral ∫ (4 / (cosx^2) - 3cosx +2 / (x-1)) dxresheniye. ∫ (4 / (cosx^2) - 3cosx +2 / (x-1)) dx = 4dx / (cosx^2) - 3∫cosxdx +2dx / (x-1) =4tgx-3sinx+2ln |x-1| + C.
3. Now it is possible to consider the rule which allows to expand possibilities of use of the table of the main integrals. If ∫f(x)dx=F(x) + C, then ∫f(kx+b) of dx=(1/k) F(kx+b) +SPrimer 2. ∫sin(5x)dx=-(1/5)cos(5x)+ C.
4. 2. Decomposition of subintegral function. This reception consists in transformation of subintegral function, using formulas of algebra and trigonometry. Subintegral function is presented in the form of the sum of functions from which integrals can be taken easily. Example 3. ∫ (1+ (cosx) ^2 / (1+cos(2x)) dx= [1+cos(2x) =2 (cosx) ^2]= ∫ (1+ (cosx) ^2/2(cosx) ^2) dx == (1/2) 1 / (cosx) ^2) dx+(1/2) ∫dx= (1/2) (tgx+x) +S.Primer 4. ∫dx / (sinx) ^2) (cosx) ^2)) = ∫ ((sinx) ^2+(cosx) ^2) / ((sinx) ^2) (cosx) ^2)) dx= ∫ (1 / (cosx) ^2+1 / (sinx) ^2) dx=tgx-ctgx+C.
5. 3. Leading under the sign of differential. This reception is based on property of invariancy of formulas of integration. Subintegral function will be transformed to a type of f (u(x))u’ (x), and then u factor’ (x) brought under the sign of differential integrated) – u’ (x) dx=d (u(x)) then the formula ∫ (f (u(x)) du(x)) by =u (x) + C is applied.
6. Example 5. ∫ (arctgx / (1+x^2)) of dx= | dx / (1+x^2) =d (arctgx) |= ∫ (arctgxd(arctgx))= (1/2) (arctgx) ^2+C.Пример 6. ∫xsqrt(1-x^2) dx= | d (1-x^2)=-2xdx |=-(1/2) ∫ ((1-x^2) ^ (1/2+1)) / (1/2+1) + C =-(1/3) sqrt ((1-x^2) ^3) + S.Primer 7. ∫ ((cosx) ^3) sin(2x)dx=2 ∫ (cosx) ^3) cosxsinxdx=-2 ∫ (cosx) ^4) d(cosx)=-(2/5)(cosx) ^5+C.