Revision: Calculus >> Integrals Maths Commerce (English Medium) Class 12 CBSE

The collection of all anti-derivatives of a function is called its indefinite integral.

Here: f(x) = integrand, dx = variable of integration, C = constant of integration.

A definite integral is connected with finding the area under a curve over a given interval. The chapter introduction presents area as one of the central motivating ideas behind integral calculus.

If the derivative of a function F(x) is f(x), then F(x) is called an antiderivative or integral of f(x). The set of all such antiderivatives is written as:

where C is an arbitrary constant called the constant of integration.

Integration by substitution is a method in which we replace a part of the integral by a new variable to simplify the integration.

General Formula:

If \[x = g(t), \ dx = g'(t) dt\] then \[\int f(x) dx = \int f(g(t))g'(t) dt\]

Integration using trigonometric identities means converting a trigonometric expression into an easier form with the help of standard identities before integrating.

Proper Rational Function

A rational function \[ \frac{P(x)}{Q(x)} \] is called proper if \[ \deg P(x) < \deg Q(x). \]

Improper Rational Function

It is called improper if \[ \deg P(x) \geq \deg Q(x). \]

An improper rational function must first be reduced to a proper rational function by long division.

Integration by partial fractions is a method used to integrate rational functions, that is, functions of the form

\[\frac{p(x)}{q(x)}\], where both numerator and denominator are polynomials.

A proper rational function can be expressed as a sum of simpler rational functions called partial fractions.

A rational function is a function of the form \[ \frac{P(x)}{Q(x)} \] where P(x) and Q(x) are polynomials and Q(x) \neq 0.

Integration by parts is a method of integration based on the product rule of differentiation.

If u and v are differentiable functions, then \[ \boxed{\int u\, dv = uv - \int v\, du} \]

For a product f(x)g(x), \[ \boxed{\int f(x)g(x)\, dx = f(x)\int g(x)\, dx - \int\left[f'(x)\int g(x)\, dx\right]dx} \]

A definite integral represents the value of a function accumulated between two limits.
It can also be interpreted geometrically as the net area between the graph of y = f(x) and the x-axis from x = a to x = b.

If a function f is continuous on an interval, the area function is defined by

This means that A(x) gives the area accumulated from x = a to a variable point x.

If f(x) is a continuous function defined on an interval [a, b] and if Φ(x) is the antiderivative of f(x), i.e., \[\frac{d}{dx}[\phi(x)]=f(x)\] then the definite integral of f(x) over [a, b] denoted by \[\int_{a}^{b}f(x)dx\]  is defined as

\[\int_{a}^{b}f(x)dx=
\begin{bmatrix}
\phi\left(x\right)
\end{bmatrix}_{a}^{b}=\phi\left(b\right)-\phi\left(a\right)\]

\[\mathrm{If~}\frac{d}{dx}[F(x)]=f(x),\mathrm{~then~}\int f(x)dx=F(x)\]

Integration is the inverse process of differentiation.

\[\int f(x)dx=F(x)+c\]

• The arbitrary constant 'c' is called the constant of integration.
• F(x) + c is called the indefinite integral.

• \[\int \frac{dx}{x^2 - a^2} = \frac{1}{2a} \log \left| \frac{x - a}{x + a} \right| + C\]
• \[\int \frac{dx}{a^2 - x^2} = \frac{1}{2a} \log \left| \frac{a + x}{a - x} \right| + C\]
• \[\int \frac{dx}{x^2 + a^2} = \frac{1}{a} \tan^{-1} \left(\frac{x}{a}\right) + C\]
• \[\int \frac{dx}{\sqrt{x^2 - a^2}} = \log \left| x + \sqrt{x^2 - a^2} \right| + C\]
• \[\int \frac{dx}{\sqrt{a^2 - x^2}} = \sin^{-1} \left(\frac{x}{a}\right) + C\]
• \[\int \frac{dx}{\sqrt{x^2 + a^2}} = \log \left| x + \sqrt{x^2 + a^2} \right| + C\]

1. \[\int[f(x)]^nf^{\prime}(x)dx=\frac{[f(x)]^{n+1}}{n+1}+c\quad(n\neq-1)\]

2. \[\int\frac{f^{\prime}(x)}{f(x)}dx=\log|f(x)|+c\]

\[\int e^x\left[\left.f(x)+f^{\prime}(x)\right.\right]dx=e^xf(x)+c\]

If, u = f(x) ⇒ \[\frac{du}{dx}=f^{\prime}(x)\]

then \[\int[f(x)]^nf^{\prime}(x)dx=\frac{[f(x)]^{n+1}}{n+1}+c\quad(n\neq-1)\]

Linear Substitution Rule:

If $b$, then

\[\int(ax+b)^ndx=\frac{(ax+b)^{n+1}}{a(n+1)}+c\quad(n\neq-1)\]

Statement:

If f(x) and g(x) are any two differentiable functions of x and G(x) is the antiderivative of g(x), i.e., \[G(x)=\int g(x)dx\]. Then 

\[\int f(x)g(x)dx=f(x)G(x)-\int f^{\prime}(x)G(x)dx\]

(A) Non-repeated linear factors

\[\frac{Ax+B}{(x-a)(x-b)}=\frac{A}{x-a}+\frac{B}{x-b}\]

(B) Repeated linear factor

\[\frac{Ax+B}{(x-a)^n}=\frac{A_1}{x-a}+\frac{A_2}{(x-a)^2}+\cdots+\frac{A_n}{(x-a)^n}\]

(C) Quadratic factor (not factorisable)

\[\frac{Ax+B}{ax^2+bx+c}\]

1.\[\int\sqrt{(a^{2}-x^{2})}dx=\frac{1}{2}x\sqrt{(a^{2}-x^{2})}+\frac{1}{2}a^{2}\sin^{-1}\left(\frac{x}{a}\right)+c\]

2. \[\int\left(\sqrt{a^{2}+x^{2}}\right)dx=\frac{1}{2}x\sqrt{(a^{2}+x^{2})}+\frac{1}{2}a^{2}\log|x+\sqrt{(a^{2}+x^{2})}|+c\]

Let I = `int sqrt(a^2 - x^2) dx`

= `int sqrt(a^2 - x^2)*1 dx`

= `sqrt(a^2 - x^2)* int 1 dx - int [d/dx (sqrt(a^2 - x^2))* int 1 dx]dx`

= `sqrt(a^2 - x^2)*x - int [1/(2sqrt(a^2 - x^2))*d/dx (a^2 - x^2)*x]dx`

= `sqrt(a^2 - x^2)*x - int 1/(2sqrt(a^2 - x^2))(0 - 2x)*x  dx`

= `sqrt(a^2 - x^2)*x - int (-x)/sqrt(a^2 - x^2)*x  dx`

= `xsqrt(a^2 - x^2) - int (a^2 - x^2 - a^2)/sqrt(a^2 - x^2)dx`

= `xsqrt(a^2 - x^2) - int sqrt(a^2 - x^2)dx + a^2 int dx/sqrt(a^2 - x^2)`

= `xsqrt(a^2 - x^2) - I + a^2sin^-1(x/a) + c_1`

∴ 2I = `xsqrt(a^2 - x^2) + a^2sin^-1(x/a) + c_1`

∴ I = `x/2 sqrt(a^2 - x^2) + a^2/2 sin^-1(x/a) + c_1/2`

∴ `int sqrt(a^2 - x^2)dx = x/2 sqrt(a^2 - x^2) + a^2/2sin^-1(x/a) + c`, where `c = c_1/2`.

If f is continuous on [a, b] and

If f is continuous on [a, b] and F is any antiderivative of f, then

This is the formula most often used in exams to evaluate definite integrals.

Theorem 1:

 Let f be a continuous function on the closed interval [a, b] and let A (x) be the area function. Then A′(x) = f (x), for all x ∈ [a, b]

Theorem 2:

Let f  be a continuous function defined on the closed interval [a, b], and F be an antiderivative of f. Then  \[\int_a^bf(x)dx=\left[\mathbf{F}(x)\right]_a^b=\mathbf{F}(b)-\mathbf{F}(a)\]

• Primitive
  Another name for anti-derivative.
• Indefinite Integral
  The family of all anti-derivatives of a function.
• Definite Integral
  An integral taken between two fixed limits, commonly used to represent area or total accumulation.
• Integral Calculus
  The branch of calculus dealing with anti-derivatives, accumulation, and areas under curves.

• Integration is the inverse process of differentiation.

• The result of indefinite integration is called the antiderivative or primitive.

• The indefinite integral is written as: ∫f(x) dx = F(x) + C.

• The constant C must always be added in indefinite integrals.

• Integration by substitution is the reverse process of the chain rule.

• Choose the substitution so that the integral becomes simpler, not more complicated.

• Always rewrite both the function and \(dx\) in terms of the new variable.

• For indefinite integrals, back-substitute and add \(C\).

• For definite integrals, limits should also be changed if the solution is continued in the new variable.

• Trigonometric substitution is mainly used for radicals involving \(a^2-x^2\), \(x^2+a^2\), and \(x^2-a^2\).

• First inspect the pattern in the integrand.

• Do not integrate complicated trigonometric expressions directly if an identity can simplify them first.

• After simplification, integrate term by term carefully.

• Always add the constant of integration, \(C\).

• First check whether the rational function is proper or improper.

• Use long division before decomposition if the fraction is improper.

• Factorise the denominator completely before choosing partial fractions.

• For each distinct linear factor, use a constant numerator such as A, B, or C.

• For a repeated linear factor, include every power separately.

• For an irreducible quadratic factor, use a linear numerator of the form Bx + C.

• After decomposition, integrate each term separately using standard formulas.

• Formula:

  \[\int u dv = uv - \int v du\]

• Choose u by LIATE

• For log x and inverse trig, multiply by 1

• Repeated parts may be needed for \[e^x \sin x\], \[e^x \cos x\].

• Convert the integrand into a known standard form before integrating.

• For \[x^2 - a^2\], factorize and use partial fractions.

• For \[x^2 + a^2\], the answer usually involves \[\tan^{-1}\].

• For \[\sqrt{a^2 - x^2}\], the answer usually involves \[\sin^{-1}\].

• For general quadratics, complete the square first.

• For \[px + q\] in the numerator, relate it to the derivative of the denominator.

• Always write the constant of integration C in the final answer.

• Used to find exact accumulated value over a fixed interval.

• Written as \[\int_{a}^{b} f(x) \, dx\].

• Evaluated using \[F(b) - F(a)\].

• Gives a unique numerical value.

• Represents net area geometrically.

• The theorem connects differentiation and integration.

• If \[A(x) = \int_{a}^{x} f(t) \, dt\], then \[A'(x) = f(x)\].

• If  F'(x) = f(x), then \[\int_{a}^{b} f(x) \, dx = F(b) - F(a)\].

• The result is used to evaluate definite integrals quickly.

• The function should be continuous on the interval for direct use of the theorem.

• Look for an inner function and its derivative.

• Choose substitution carefully.

• If the integral is continued in the new variable, change the limits accordingly.

• Integrate in the new variable.

• Do not add +C in a definite integral.

• Once the limits have been changed to the new variable, there is no need to substitute back to the original variable.

For choosing the first function:

L I A T E

• Logarithmic

• Inverse trigonometric

• Algebraic

• Trigonometric

• Exponential

1.\[\frac{d}{dx}{\left[\int f(x)dx\right]}=f(x)\]

2. \[\int cf(x)dx=c\int f(x)dx\]

3. \[\int(u+v-w)dx=\int udx+\int vdx-\int wdx\]