This chapter covers the concept of integrals, including indefinite and definite integrals, crucial for calculating areas under curves and solving practical problems in various fields.
Integrals – Formula & Equation Sheet
Essential formulas and equations from the Integrals chapter of Mathematics Part - II, tailored for Class 12 in Mathematics.
This one-pager compiles key formulas and equations from the Integrals chapter of Mathematics Part - II. Ideal for exam prep, quick reference, and solving time-bound numerical problems accurately.
Key concepts & formulas
Essential formulas, key terms, and important concepts for quick reference and revision.
Formulas
∫ x^n dx = (x^(n+1))/(n+1) + C, n ≠ -1
This is the formula for integrating power functions, where n is a real number and C is the constant of integration.
∫ sin(x) dx = -cos(x) + C
The integral of sine function gives cosine function with a negative sign, plus a constant of integration.
∫ cos(x) dx = sin(x) + C
The integral of cosine function yields sine function plus a constant.
∫ e^x dx = e^x + C
The integral of the exponential function e^x is itself plus a constant.
∫ 1/x dx = log|x| + C
The integral of the reciprocal function results in the natural logarithm of the absolute value of x plus a constant.
∫ sec^2(x) dx = tan(x) + C
The integral of secant squared function equals tangent function plus a constant.
∫ csc^2(x) dx = -cot(x) + C
The integral of cosecant squared function equals negative cotangent function plus a constant.
∫ f'(x) dx = f(x) + C
The integral of the derivative of a function essentially yields the original function plus a constant.
∫ a * f(x) dx = a * ∫ f(x) dx
Integrating a constant multiplied by a function can be simplified by factoring the constant out of the integral.
∫ [f(x) + g(x)] dx = ∫ f(x) dx + ∫ g(x) dx
The integral of the sum of two functions is the sum of the integrals of each function.
Equations
∫ (sin x cos x) dx = (1/2)sin^2 x + C
The integral of sin x cos x can be found using the identity sin(2x) = 2sin x cos x.
∫ (a^2 - x^2)^(1/2) dx = (1/2)(x * (a^2 - x^2)^(1/2) + a^2 * arcsin(x/a)) + C
This formula represents the integral for a function resembling the area of a circle.
∫ f(x)g'(x) dx = f(x)g(x) - ∫ f'(x)g(x) dx
This is integration by parts formula, which helps in integrating products of functions.
∫ (dx/(x^2 + a^2)) = (1/a)arctan(x/a) + C
This formula gives the integral of a rational function related to the arctangent function.
∫ (1/(x^2 + a^2)) dx = (1/a)arctan(x/a) + C
This integral shows the relationship of arctan with a specific rational function.
∫ sec(x) dx = ln |sec(x) + tan(x)| + C
This formula yields the integral of the secant function utilizing logarithmic properties.
∫ csc(x) dx = ln |csc(x) - cot(x)| + C
The integral of the cosecant function involves logarithmic identities.
∫ 1/(a + bx) dx = (1/b)ln|a + bx| + C
Integrating a linear function in the denominator results in a logarithmic function.
∫ f(g(x))g'(x) dx = F(g(x)) + C
This represents the substitution method, allowing integration of composite functions.
∫ dx/(x^n) = (1/n-1)x^(1-n) + C, n ≠ 1
Integrating a power function that is not equal to 1 requires division by the new exponent.
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