Properties of indefinite integral
1. The derivative (the differential) of an indefinite integral is equal to the function (to the expression) under the integral sign:
Corollary.Correctness of integration can by tested by differentiation.
2. The indefinite integral of the derivative (of the differential) of a function equals the sum of this function and an arbitrary constant:
Corollary.A function can be recovered from its derivative or differential with accuracy to additive constant.
Ex. 4. .
3 (additivity). The indefinite integral of an algebraic sum of a finite number of function equals the sum of their integrals, for ex.
■It’s sufficient to prove that the derivatives of the left and right sides of the equality are equal. But by the property 1
4 (homogeneity). A constant factor can be taken out of the integral sign:
Prove the property yourselves.
Corollary 1(linearity). For any functions and constants
Corollary 2.On the base of the linear property and the table of simplest integrals one often can fulfil so-called direct integration.
Ex. 8. On the base of Ex. 5-7
POINT 3. INTEGRATION BY SUBSTITUTION
Theorem 2.Let functions are continuous in corresponding intervals and the function has the continuously differentiable inverse function . In this case the next formula (formula of change of variable) is true
( 4 )
The formula implies returning to preceding variable x after integration with respect to the variable t. The word “differentiation” always means finding of differentials.
■The first method. The derivatives of the left and right sides of the formula (4) are equal because of
The second method. If is a primitive of the function , then the function is the primitive of the function on the strength of
Therefore by virtue of definition of the indefinite integral
Note 3.The formula (4) is often applied “from the right to the left”, and in this case it’s useful to write it in the next form
. ( 5 )
The formula (5) means: if an integrand is represented as a product of some function f of a function and the derivative of this latter then it’s well to put .
Note 4.We can use any letter instead t in the formula (5).
Ex. 10. Prove yourselves that .
Ex. 12. Evaluate the indefinite integral .
The integrand is a product of a function of , namely , and the derivative of (up to a constant factor , for ). On the base of the formula (5) we can put or better . We suppose that and so . Therefore
Ex. 13. Calculate the indefinite integral .
The integrand is a product of the function of and (up to a factor -20) the derivative of . So, putting , we reduce the given integral to a tabular one (see the tabular formula (12) where we must take )
Ex. 14. The case when an integrand is a fraction, the numerator of which is the derivative of the denominator. The integral is reduced to a tabular one (see 2)
( 6 )
For ex. a) ( 7 )
b) ( 8 )
Ex. 15. Prove that ( 9 )
Solution. Using the method of Ex. 12 we put . Hence, .
Ex.16. Indefinite integrals
( 10 )
are reduced to sums of integrals of the types (8), (9) and those tabular with the help of the substitution
. ( 11 )
For example: a) Using the substitution (11), the formula (8) and the tabular integral No. 13, we get
b) With the help of (11), (9) and the tabular integral No. 12 we obtain
Ex. 17. To evaluate the indefinite integral
we put , whence (with the help of the tabular integral No. 13)
Ex. 18. To prove the tabular formula No. 14 we’ll use so-called Euler’s substitution