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.

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

If two functions are written in the form uu and dvdv, then integration by parts is based on the product rule of differentiation.

\[\int\left(\mathrm{u.v}\right)\mathrm{dx}=\mathrm{u}\int\mathrm{v}\mathrm{dx}-\int\left(\frac{\mathrm{du}}{\mathrm{dx}}\right).\left(\int\mathrm{v}\mathrm{dx}\right)\mathrm{dx}\]

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.

• General form: ∫f(x) dx = F(x) + C.

• The constant CC 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.

• Change the limits immediately.

• Integrate in the new variable.

• Do not add +C in a definite integral.

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