Integration Formulas for Class 12

Students in early grades learn how to determine the areas of various geometrical shapes using some elementary geometry methods. These methods and formulas are fundamental to learning multiple applications of mathematics in real-life problems. However, to determine the areas covered under a curve we need integration formulas. Integration formulas for class 12 are useful in determining the area covered under a curve and between two curves. These formulas are important to solve a variety of questions based on integration by using different methods. Apart from mathematics, these integration formulas for class 12 are applied to solve scientific and engineering problems.

List of Integration Formulas for Class 12

Here is the list of integration formulas for class 12:

• ∫ f(x) dx = F(x) + C
• Power Rule: ∫ xⁿ dx = (xⁿ⁺¹)/ (n+1)+ C. (Where n ≠ -1)
• Exponential Rules: ∫ e^x dx = e^x + C

∫ a^x dx = a^x/ln(a) + C

∫ ln(x) dx = x ln(x) -x + C

• Constant Multiplication Rule: ∫ a dx = ax + C, where a is the constant.
• Reciprocal Rule: ∫ (1/x) dx = ln(x)+ C
• Sum and Difference Rules:

∫ [f(x) + g(x)] dx = ∫f(x) dx + ∫g(x) dx

∫ [f(x) - g(x)] dx = ∫f(x) dx - ∫g(x) dx

• ∫k f(x) dx = k ∫f(x) dx, , where k is any real number.
• Integration by parts: ∫ f(x) g(x) dx = f(x) ∫ g(x) dx - ∫[d/dx f(x) * ∫ g(x) dx]dx
• ∫cos x dx = sin x + C
• ∫ sin x dx = -cos x + C

• ∫ sec² x dx = tan x + C
• ∫ cosec² x dx = -cot x + C
• ∫ sec x tan x dx = sec x + C
• ∫ cosec x cot x dx = - cosec x + C
• Area of the region bounded by the curve y = f (x), x-axis and the lines x = a and x = b (b > a) : a∫b y dx = a∫bf(x) dx.
• Area of the region bounded by the curve x = φ (y), y-axis and the lines y = c, y = d : c∫d x dy = c∫d φ (y) dy.
• Area of the region enclosed between two curves y = f (x), y = g (x) and the lines x = a, x = b : b∫a[f(x) - g(x)] dx

Applications of Integration Formulas for Class 12

Integration formulas for class 12 are applied in mathematics to determine areas, volumes, central points and many other useful aspects related to a non-uniform curve with some given function. Here are a few examples of real-life applications of these integration formulas for class 12.

• Integration formulas for class 12 are used in physics to calculate the center of gravity, mass and momentum of inertia of vehicles.
• These formulas are also applied to calculate the mass and momentum of satellites or a tower.
• The application of these formulas can be found in finding the velocity and trajectory of a satellite at the time of placing it in orbit.

Tips to Memorize Integration formulas for class 12

The following tips can be helpful for the students to remember the integration formulas for class 12:

• Students must have a clear understanding of the underlying logic of all formulas for easing the memorization process.

• It is better to clarify all doubts before trying to memorize integration formulas for class 12. Eliminating all doubts at the start will help in memorizing these formulas better.

• Students should try to solve several problems based on integration formulas for class 12 to get familiarized with the usage of these formulas in different scenarios. This will allow students to recall them quickly when required.

• Students can use various resources like formula sheets available on the internet to revise these integration formulas for class 12.

Integration Formulas for Class 12 Examples

Example: Find the integral of Sin2x.

Solution: We know that, d/dx(Cos2x) = - 2Sin2x

⇒ Sin2x = -½ d/dx(Cos2x)

⇒ Sin2x = d/dx(-½ Cos2x)

Thus the integral or anti-derivative of Sin2x = -½ Cos2x.

Students can download the printable Maths Formulas class 12 sheet from below.