List of Integration Formulas: In Class 12 Maths, integration is the inverse process of differentiation, also known as Inverse Differentiation. It is a method of calculating the total value by adding up several components. It is the process of determining a function with its derivative. Integration formulas can integrate algebraic equations, trigonometric ratios, inverse trigonometric functions, logarithmic and exponential functions, and other functions.

Integration Formulas for Class 12 are used to determine a function’s antiderivative. We obtain a family of functions in I if we differentiate a function f in an interval I. If we know the values of the functions in I, we can calculate the function f. This inverse differentiation procedure is known as integration. Scroll down to check & download Integration Formula List PDF from this article.

Integration Formulas: Symbols and Meanings

Before providing you with the formula list, we have tabulated all the important symbols, terms, and phrases used in integration and what they mean:

List of Integration Formulas for Class 12

The integral of a function f(x)f(x) with respect to xx is written as∫f(x)dx. The basic formulas used commonly in integrations are listed below:

Basic Integration Formula List:

Some generalised results obtained using the fundamental theorems of integrals are remembered as integration formulas in indefinite integration. Below are the Integration basic formulas for your reference:

• ∫ xⁿ.dx = x^{(n + 1)}/(n + 1)+ C
• ∫ 1.dx = x + C
• ∫1/x.dx = log|x| + C
• ∫ e^x.dx = e^x + C
• ∫ a^x.dx = a^x /loga+ C
• ∫ e^x[f(x) + f'(x)].dx = e^x.f(x) + C

Trigonometric Integration Formulas

A list of formulas on trigonometric function is given below:

• ∫ cosx.dx = sinx + C
• ∫ sinx.dx = -cosx + C
• ∫ cosec²x.dx = -cotx + C 
• ∫ sec²x.dx = tanx + C
• ∫ cosecx.cotx.dx = -cosecx + C 
• ∫ secx.tanx.dx = secx + C 
• ∫ tanx.dx =log|secx| + C 
• ∫ cotx.dx = log|sinx| + C 
• ∫ cosecx.dx = log|cosecx – cotx| + C
• ∫ secx.dx = log|secx + tanx| + C 

Formulas of Inverse Trigonometric Integration Functions

Here is the list of all important formulas on inverse trigonometric functions:

• ∫1/√(1 – x²).dx = sin^-1x + C
• ∫ /1(1 – x²).dx = -cos^-1x + C
• ∫1/(1 + x²).dx = tan^-1x + C
• ∫ 1/(1 +x² ).dx = -cot^-1x + C
• ∫ 1/x√(x² – 1).dx = -cosec^-1 x + C
• ∫ 1/x√(x² – 1).dx = sec^-1x + C

Advanced Formulas on Integration

Here is the list of some important and most commonly asked formulas on advanced integration functions:

• ∫ 1/(a² – x²).dx =1/2a.log|(a + x)(a – x)| + C
• ∫1/(x² – a²).dx = 1/2a.log|(x – a)(x + a| + C 
• ∫1/(x² + a²).dx = 1/a.tan^-1x/a + C
• ∫1/√(x² – a²)dx = log|x +√(x² – a²)| + C
• ∫1/√(a² – x²).dx = sin^-1 x/a + C 
• ∫ √(x² – a²).dx =1/2.x.√(x² – a²)-a²/2 log|x + √(x² – a²)| + C 
• ∫√(a² – x²).dx = 1/2.x.√(a² – x²).dx + a²/2.sin-1 x/a + C
• ∫1/√(x² + a² ).dx = log|x + √(x² + a²)| + C 
• ∫ √(x² + a² ).dx =1/2.x.√(x² + a² )+ a²/2 . log|x + √(x² + a² )| + C

Different Integration Formulas

Three types of integration methods are generally used: Integration by parts formula, Integration by Substitution formula and Integration by partial fractions formula. Let us look at each of these formulas on integration, one by one.

Integration by Parts Formula

When any given function is a product of two different functions, the integration by parts formula or partial integration can be applied to evaluate the integral. The integration formula using partial integration methos is as follows:

∫ f(x).g(x) = f(x).∫g(x).dx -∫(∫g(x).dx.f'(x)).dx  + c

For instance: ∫ xe^x dx is of the form ∫ f(x).g(x). Therefore, we must apply the appropriate integration formula and evaluate the integral accordingly.

f(x) = x and g(x) = e^x

Thus ∫ xe^x dx = x∫e^x .dx  – ∫( ∫e^x .dx. x). dx+ c

= xe^x – e^x + c

Integration by Substitution Formula

If a given function is a function of another function, we can apply the integration formula for substitution to solve that integral. For instance, if
I = ∫ f(x) dx,
where
x = g(t) so that dx/dt = g'(t), then we write dx = g'(t) 
Take for instance
I = ∫ f(x) dx = ∫ f(g(t)) g'(t) dt
For example: Consider ∫ (3x +2)⁴ dx
The integration formula of substitution is given as follows.
Take u = (3x+2). ⇒ du = 3. dx
Thus ∫ (3x +2)⁴ dx =1/3. ∫(u)⁴. du
= 1/3. u⁵ /5 = u⁵ /15
= (3x+2)⁵ /15

Partial Fractions Integrating Formula

To find the integral of an improper fraction like P(x)/Q(x), in which the degree of P(x) < that of Q(x), we can use integration by partial fractions. In this method, we split the fraction using partial fraction decomposition as P(x)/Q(x) = T(x) + P11 (x)/ Q(x), in which T(x) is a polynomial in x and P11 (x)/ Q(x) is a proper rational function.

Assume that A, B and C are real numbers, we can have the following types of simpler partial fractions associated with various types of rational functions.

Rational Fractions	Partial Fractions
(px + q)/(x-a)(x – b)	A/(x – a) + B/ (x-b)
(px + q)/(x-a)n	A1/(x-a) + A2/(x-a)2 + ………. An/(x-a)n
(px2 + qx + r)/(ax2 + bx + c)n	(A1x + B1)/(ax2 + bx + c) + (A2x + B2)/(ax2 + bx + c)2 + …(Anx + Bn)/(ax2 + bx + c)n
(px2 + qx + r)/(ax2 + bx + c)	(Ax + B)/(ax2 + bx + c)
(px2 + qx + r)/(x-a)(x-b)(x-c)	A/(x – a) + B/ (x-b) + C/ (x-c)
(px2 + qx + r)/(x2 +bx +c)	A/(x-a) +(Bx+C)/(x2 +bx +c)

For Example: ∫ 3x+7/ x² -3x + 2

Upon resolving it into partial fractions, we get

3x+7/ x² -3x + 2 = A/(x-2) + B/ (x-1)

= A(x-1) + B(x-2)/ (x-2)(x-1)

Equating the numerators, we get 3x +7 = A(x-1)+B(x-2)

Find B by giving x = 1⇒ 10 = B

Find A by giving x = 2⇒ 13 = A

Thus 3x+7/ x² -3x + 2 = 13/(x-2) + 10(x-1)

Applying the integration formula, we get

∫ (3x+7/ x² -3x + 2) = ∫ 13/(x-2) + ∫ 10(x-1)

∫ (3x+7/ x² -3x + 2) = 13 log |x-2| – 10 log |x-1| + C

Definite Integral Formula

These are the integrations with pre-existing limit values, making the final value of the integral definite:

Indefinite Integral Formula

These are integrations that lack a pre-existing value of limits, rendering the final value of the integral limitless. C denotes the integration constant.

∫ g'(x) = g(x) + C

All Important Formulas on Integration PDF

You can have a look and also download the pdf for both integration and differentiation formulas from below:

Check out some more formulas that will aid you in your preparation.

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FAQs on Integration Formulas