26. Indefinite Integrals - Revision Facilitator

1. Indefinite Integral - Antiderivative – Primitive

2. Integrals of some standard functions

3. Integration – Some standard formulas

a. ∫kf(x)dx =

b. ∫[f(x)± g(x)]dx =

c. d/dx [∫f(x)dx] =

4. Integration by substitution

5. Integrals of the form [f '(x)/f(x)]dx

6. Integrals of the form sin ^m x cos ^n x dx

7. Integrals of the functional form 1/(x²±a²)

8. Integrals of the form [1/(ax²+bx+c)]dx

9. Integrals of the form [1/√(ax²+bx+c)]dx

10. Integrals of the form [(px+q)/(ax²+bx+c)]dx

11. Integrals of the functional form [P(x)/(ax²+bx+c)]dx

12. Integrals of the form [(px+q)/√(ax²+bx+c)]dx

13. Integrals of the functional form [1/(a sin²x + b cos²x +c)]dx

14. Integrals of the functional form [1/(a sin x + b cos x +c)]dx

15. Integrals of the functional form [(a sin x + b cos x)/(c sin x + d cos x)]dx

16. Integrals of [(a sin x+b cos x +c)/(p sin x + q cos x +r)] dx


17. Integration by parts

18. Integral of e^x [f(x)+f'(x)]dx


19. Integrals of e^ax sinbx dx,e^ax cos bx dx

20. Integrals of √(a²±x²) and √(x²-a²)

21.Integrals of the functions of the form √(ax²+bx+c)dx

22. Integrals of the functions of the form (px+q)[√(ax²+bx+c)]dx

23. Integration of Rational Algebraic Functions by Using Partial Fractions

24. Integration of [(x²+1)/(x^4+λx²+1)]dx

25. Integration of Function [G(x)/(P√Q)]dx

Indefinite Integral - Antiderivative - Primitive

a function ф(x) is called a primitive or an antiderivative of a function f(x) if ф'(x) = f(x).

For a function f(x), the collection of all its primitives is called the indefinite integral of f(x0 and is denoted by ∫f(x)dx.


∫f(x)dx = ф(x)+C (where C is a constant)

Here ∫ is the integral sign, f(x) is th integrand, x is the variable of integration and dx is the element of integration or differential of x.

The process of finding an indefinite integral of a given function is called integration of the function.

Integrals of some standard functions

Basic integrals

Add C (constant) to given ∫f(x)dx

S.no. f(x) ∫f(x)dx

1. 0 ... C (constant)

2. xⁿ (n not equal to -1) ... xⁿ⁺¹/(n+1)

3. 1/x ... ln|x|

4. e^x ... e^x

5. a^x ... a^x/ln a

Trigonometric functions

6. sin x ... -cos x

7. cos x ... sin x

8. Cosec²x ...-cot x

9. sec²x ... tan x

Integration - Some Standard Results

1. ∫kf(x)dx = k∫f(x)dx

2. ∫[f(x)± g(x)]dx = ∫f(x)dx ± ∫g(x)dx

3. d/dx [∫f(x)dx] = f(x)

Integration by substitution

If ф(x0 si a ocntinuously differntiable function, then to solve

∫f(ф(x))ф'(x)dx; we substitute ф(x) = t and ф'(x)dx will be equal to dt.

Hence the problem is transformed to ∫f(t)dt

Integrals of the form [f'(x)/f(x)]dx

∫[f'(x)/f(x)]dx = log[f(x)]

Using the formula

∫tan x dx = ∫(sin x/cos x)dx

If f(x) = t, f'(x)dx = dt
cos x = t;
-sin x dx = dt
sin x dx = -dt

∫(sin x/cos x)dx = ∫-dt/t = -log |t|+c = - log|cos x|+C
= log |sec x|+C

Integrals of the form sin ^m x cos ^n x dx

if m power of sin x is odd, put cosx = t.

If n power of cos x is odd, put sin x = t.

If both m, and n are odd use De'Moivre's theorem.

Integrals of the form [1/(x²±a²)]dx

∫(1/(x²+a²)dx = (1/a)tanˉ¹(x/a) + C


∫(1/(x²-a²)dx = (1/2a)log|(x-a)/(x+a)|+C

Integrals of the form [1/(ax²+bx+c)]dx

Make the coefficient of x² as unity. divide by a,

Add and subtract square of half of the coefficient of x to the expression.

Integrals of the form [1/√(ax²+bx+c)]dx

Make the coefficient of x² as unity. divide by a,

Add and subtract square of half of the coefficient of x to the expression.

Integrals of the form [(px+q)/(ax²+bx+c)]dx

Express numerator as
(px+q) = λ(derivative of denominator) + µ

Integrals of the functional form [P(x)/(ax²+bx+c)]dx

P(x) is a polynomial.

Divide P(x) by the denominator to Q(x)+R(x)/(ax²+bx+c)

R(x) will be linear or first degree equation.

Integrals of the form [(px+q)/√(ax²+bx+c)]dx

Express numerator as

px + q = λ(derivative of denominator) + µ = λ(2ax+b)+µ

Integrals of the functional form [1/(a sin²x + b cos²x +c)]dx

Divide numerator and denominator by cos²x

Replace sec²x by (1 + tan² x)

Put tan x = t

dt = sec²xdx

The integral reduces to ∫[1/(At² +Bt +C)]dt

Integrals of the functional form [1/(a sin x + b cos x +c)]dx

Put sin x = (2 tan x/2)/(1 + tan² (x/2))

cos x = (1 - tan² (x/2))/(1 + tan² (x/2))

Replace (1 + tan² (x/2)) by sec²(x/2)

Put tan (x/2) = t

dt = 1/2 sec²(x/2)dx

The integral reduces to ∫[1/(at² +bt +c)]dt

Integrals of the functional form [(a sin x + b cos x)/(c sin x + d cos x)]dx

Express numerator as

Numerator = λ(derivative of denominator) + µ(denominator)

Integrals of [(a sin x+b cos x +c)/(p sin x + q cos x +r)] dx

Express the numerator as

λ(denominator) + µ(Differential of denominator) + υ

The solution will come as λx + µ log|denominator| + υ∫dx/(p sin x + q cos x +r)

Integration by parts

∫uv dx = u(∫vdx)-∫[du/dx∫vdx]dx

Integral of e^x [f(x)+f'(x)]dx

∫e^x[f(x)+f '(x)]dx = e^xf(x)+ C

Integrals of e^ax sinbx dx,e^ax cos bx dx

∫e^ax sinbx dx = [e^ax/(a²+b²)[[a sin bx - b cos bx) +C

∫e^ax cos bx dx = [e^ax/(a²+b²)[[a cos bx + b sin bx) +C