Integral Calculus Formula
Integration is the process of combining the parts to form a whole. Here the integral calculus formula is used to find the area enclosed by the given equation of the curve, with the coordinate axes. The integral calculus formulas include formulas of integration, definite integrals, and differential equations.
What is the Integral Calculus Formula?
The following three formulas are the three broad set of formulas on integrals, definite integrals, and differential equations.The definite integrals are similar to the integrals, but include the boundaries with upper limit and lower limit for the functions.
Formula 1
Integral formulas can be applied to basic algebraic expressions, trigonometric ratios, logarithmic functions, and exponential terms.
\(\int x^n.dx = \dfrac{x^{n + 1}}{n + 1} + C \)
\(\int 1.dx = x + C \)
\(\int e^x.dx = e^x + C \)
\(\int\frac{1}{x}.dx = Logx + C \)
\(\int a^x.dx = \dfrac{a^x}{loga} + C \)
\(\int Cosx.dx = Sinx + C \)
\(\int Sinx.dx = Cosx + C \)
\(\int Sec^2x.dx = Tanx + C \)
\(\int Cosec^2x.dx = Cotx + C \)
\(\int Secx.Tanx.dx = Secx + C \)
\(\int Cosecx.Cotx.dx = Cosecx + C \)
Formula 2
Definite integral formulas can be applied to the same integral formulas, but with a lower and upper limit. The answer for a definite integral problem is a numeric value and it calculates the area of the curve within these limits.

\(\int^b_a f'(x).dx = f(b)  f(a)\)

\(\int^b_a f(x).dx = \int^b_a f(t).dt\)

\(\int^b_a f(x).dx = \int^a_b f(x).dx\)

\(\int^b_a f(x).dx = \int^b_c f(x).dx + \int^c_a f(x).dx\)

\(\int^b_a f(x).dx = \int^b_a f(a + b x).dx \)

\(\int^a_0 f(x).dx = \int^a_0 f(a x).dx \)

\(\int^{2a}_0 f(x).dx = 2\int^a_0 f(x).dx \)

\(\int^a_{a} f(x).dx = 2\int^a_0 f(x).dx \), f is an even function.

\(\int^a_{a} f(x).dx = 0 \) , f is an odd function.
Formula 3
Differential equations formulas is another important set of formulas of integral calculus and it applies the combined concepts of differentiation and integration. The general solution of a differential equation can be obtained through the process of integration.

Homogenous Differential Equation  f(λx, λy) = λ^{n}f(x, y)

Linear Differential Equation  dy/dx + Py = Q

General solution of Linear Differential Equation is \(y = e^{\int P.dx}.\int(Q.e^{\int P.dx}).dx + C\)
Let us check a few examples to know the applications of the integral calculus formula..
Solved Examples on Integral Calculus Formula

Example 1: Find the integral of 4x^{3 }+ 5x^{2 }+ 1/x, through the application of the above integral calculus formula.
Solution:
\(\int \) (4x^{3 }+ 5x^{2 }+ 1/x).dx = \(\int \)4x^{3}.dx + \(\int \)5x^{2}.dx + \(\int \).(1/x).dx
= 4.x^{4}/4 + 5.x^{3}/3 + logx + C
= x^{4} +5/3.x^{3} + logx + C
Answer: \(\int \) (4x^{3 }+ 5x^{2 }+ 1/x).dx = x^{4} +5/3.x^{3} + logx + C 
Example 2: Integrate Cosecx(Cosecx  Cotx).
Solution:
\(\int \) Cosecx(Cosecx  Cotx).dx
(Here applying the integral calculus formula we individually find the integrals.)
= \(\int \) Cosec^{2}x.dx  \(\int \)Cosecx.Cotx.dx
= Cotx  (Cosecx) + C
=Cotx + Cosecx + C
= Cosecx  Cotx + C
Answer: \(\int \) Cosecx(Cosecx  Cotx).dx = Cosecx  Cotx + C