Integral Calculus Formulas
Integral calculus formulas are as follows:
- d/dx {(ϕ(x))} = f(x) ⇔ ∫ f(x) dx = ϕ(x) + C
- ∫xn dx = (xn+1) / (n+1) + C, n ≠ 1
- ∫(1/x) dx = loge|x| + C
- ∫ ex dx = ex + C
- ∫ ax dx = (ax / logea) + C
- ∫ sin x dx = – cos x + C
- ∫ cos x dx = sin x + C
- ∫sec2x dx = tan x + C
- ∫ cosec2x dx = -cot x + C
- ∫sec x tan x dx = sec x + C
- ∫ cosec x cot x dx = – cosec x + C
- ∫ cot x dx = log |sin x| + C
- ∫tan x dx = – log |cos x| + C
- ∫sec x dx = log |sec x + tan x| + C
- ∫ cosec x dx = log |cosec x – cot x| + C
- ∫ [1 /√(a2 – x2)] dx = sin-1(x/a) + C
- ∫ -[1 /√(a2 – x2)] dx = cos-1(x/a) + C
- ∫ [1 / (a2 + x2)] dx = (1/a) tan-1(x/a) + C
- ∫ -[1 / (a2 + x2)] dx = (1/a) cot-1(x/a) + C
- ∫ [1 / {x√(x2 – a2)}] dx = (1/a) sec-1(x/a) + C
- ∫ -[1 / {x√(x2 – a2)}] dx = (1/a) cosec-1(x/a) + C
Let’s consider some examples for better understanding.
Example 1: Solve: ∫x5dx
Solution:
∫x5dx = [x5 + 1 / (5 + 1)] + C
Using ∫xn dx = (xn+1) / (n+1) + C, we get
∫x5dx = (x6 / 6) + C
Example 2: Find the definite integral of the function f(x) = 3x2 + 2x – 5.
Answer:
Using ∫xn dx = (xn+1) / (n+1) + C and ∫{f(x) ± g(x)} dx = ∫ f(x) dx ± ∫g(x) dx, we get
∫(3x2 + 2x – 5) dx = 3x3/3 + 2x2/2 – 5x + C
Example 3: Evaluate the indefinite integral ∫(4ex + 1/x) dx.
Answer:
As we know,
- ∫ ex dx = ex + C
- and ∫ kf(x) dx = k ∫ f(x) dx
∫ 4ex dx = 4ex + C1
and ∫ 1/x dx = log x + C2
⇒ ∫(4ex + 1/x) dx = 4ex + C1 + log x + C2
⇒ ∫(4ex + 1/x) dx = 4ex + log x + C1 + C2
⇒ ∫(4ex + 1/x) dx = 4ex + log x + + C [Where C = C1 + C2]
Integral Calculus
Integral Calculus is the branch of calculus that deals with topics related to integration. Integrals are major components of calculus and are very useful in solving various problems based on real life. Some of such problems are the Basel problem, the problem of squaring the circle, the Gaussian integral, etc. Integral Calculus is directly related to differential calculus.
This article is a brief introduction to Integral Calculus, including topics such as fundamental theorems of integral calculus, types of integral, and integral calculus formulas, definite and indefinite integrals with their properties, applications of integral calculus, and their examples.
Table of Content
- What is Integral Calculus?
- Fundamental Theorems of Integral Calculus
- Integral Definition
- Types of Integrals
- Definite Integrals
- Definite Integral Formula
- Properties of Definite Integrals
- Indefinite Integrals
- Properties of Indefinite Integrals
- Improper Integrals
- Multiple Integrals
- Integral Calculus Formulas
- Methods to Find Integrals
- Applications of Integral Calculus
- Differential vs Integral Calculus
- Integral Calculus Examples
- Practice Problems on Integral Calculus