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Definite Integration-Integration Formulas

 

Definite Integration

Integrands: f, g, u, v

Antiderivatives: F, G

Independent variables: x, t

Limits of integration: a, b, c, d

Subintervals of integration: Δxi

Arbitrary point of a subinterval: ξi

Natural numbers: n, i

Area of a curvilinear trapezoid: S

1. The definite integral of the function f(x) over the interval [a,b] is defined as the limit of the integral sum (Riemann sums) as the maximum length of the subintervals approaches zero.

∫abf(x)dx=lim\substackn→∞max Δxi→0∑i=1nf(ξi)Δxi, \int\limits_a^b f(x)dx= \lim_{\substack{ n \to \infty \\ \text{max} \ \Delta {x_i} \to 0 }} \sum_{i=1}^n f( \xi_i) \Delta x_i,
where Δxi=xi–xi–1,xi–1≤ξi≤xi. where \ \Delta x_i=x_i-x_{i-1}, x_{i-1} \le \xi_i \le x_i.

2. The definite integral of 1 is equal to the length of the interval of integration:

∫ab1dx=b–a \int\limits_a^b 1 dx= b-a

3. A constant factor can be moved across the integral sign:

∫abkf(x)dx=k∫abf(x)dx \int\limits_a^b kf(x)dx= k \int\limits_a^b f(x)dx

4. The definite integral of the sum of two functions is equal to the sum of the integrals of these functions:

∫ab[f(x)+g(x)]dx=∫abf(x)dx+∫abg(x)dx \int\limits_a^b [f(x)+g(x)]dx= \int\limits_a^b f(x)dx + \int\limits_a^b g(x)dx

5. The definite integral of the difference of two functions is equal to the difference of the integrals of these functions:

∫ab[f(x)–g(x)]dx=∫abf(x)dx–∫abg(x)dx \int\limits_a^b [f(x)-g(x)]dx= \int\limits_a^b f(x)dx – \int\limits_a^b g(x)dx

6. If the upper and lower limits of a definite integral are the same, the integral is zero:

∫abf(x)dx=0 \int\limits_a^b f(x)dx = 0

7. Reversing the limits of integration changes the sign of the definite integral:

∫abf(x)dx=–∫baf(x)dx \int\limits_a^b f(x)dx = -\int\limits_b^a f(x)dx

8. Suppose that a point c belongs to the interval [a,b]. Then the definite integral of a function f(x) over the interval [a,b] is equal to the sum of the integrals over the intervals [a,c] and [c,b]:

∫abf(x)dx=∫acf(x)dx+∫cbf(x)dx \int\limits_a^b f(x)dx= \int\limits_a^c f(x)dx + \int\limits_c^b f(x)dx

9. The definite integral of a non-negative function is always greater than or equal to zero:

∫abf(x)dx≥0 if f(x)≥0 in [a,b] \int\limits_a^b f(x)dx \ge 0 \ if \ f(x) \ge 0 \ in \ [a,b]

10. The definite integral of a non-positive function is always less than or equal to zero:

∫abf(x)dx≤0 if f(x)≤0 in [a,b] \int\limits_a^b f(x)dx \le 0 \ if \ f(x) \le 0 \ in \ [a,b]

11. Fundamental theorem of calculus

∫abf(x)dx=F(x)|ab=F(b)–F(a),if F‘(x)=f(x) \int\limits_a^b f(x)dx = F(x) |_a^b = F(b)-F(a), if \ F'(x)=f(x)

12. Substitution rule for definite integrals

if x=g(t) then if \ x=g(t) \ then
∫abf(x)dx=∫cdf(g(t))g‘(t)dt \int\limits_a^b f(x)dx = \int\limits_c^d f(g(t))g'(t)dt
where, c=g–1(a),d=g–1(b) where, \ c=g^{-1}(a), d=g^{-1}(b)

13. Integration by parts for definite integrals

∫abudv=(uv)|ab–∫abvdu \int\limits_a^b udv= (uv)|_a^b – \int\limits_a^b vdu

14. Trapezoidal approximation of a definite integral

∫abf(x)dx=b–a2nf(x0)+f(xn)+2∑i=1n–1f(xi) \int\limits_a^b f(x)dx = \frac {b-a}{2n} \left[ f(x_0) + f(x_n) + 2 \sum_{i=1}^{n-1} f(x_i) \right]

15. Approximation of a definite integral using Simpson’s rule

∫abf(x)dx=b–a3nf(x0)+4f(x1)+2f(x2)+4f(x3)+⋯+4f(xn–1)+f(xn), \int\limits_a^b f(x)dx = \frac {b-a}{3n} \left[ f(x_0) + 4f(x_1) + 2f(x_2) + 4f(x_3) + \cdots + 4f(x_{n-1}) + f(x_n) \right],
where xi=a+b–ani,i=0,1,2,⋯,n where \ x_i=a+ \frac {b-a}{n}i, i=0,1,2,\cdots ,n

16. Area under a curve

S=∫abf(x)dx=F(b)–F(a), S= \int\limits_a^b f(x)dx = F(b)-F(a),
where, F‘(x)=f(x) where, \ F'(x)=f(x)

17. Area between two curves

S=∫ab[f(x)–g(x)]dx=F(b)–G(b)–F(a)+G(a) S= \int\limits_a^b [f(x) -g(x)]dx = F(b)-G(b)-F(a)+G(a)
where, F‘(x)=f(x), G‘(x)=g(x) where, \ F'(x)=f(x), \ G'(x)=g(x)

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