Revision: Calculus >> Integrals Mathematics ISC (Science) ISC Class 12 CISCE

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.

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

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.

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.

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

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

(iii) \[\int\frac{dx}{x^{2}+a^{2}}=\frac{1}{a}\tan^{-1}\frac{x}{a}+C\]

(iv) \[\int\frac{dx}{\sqrt{x^{2}-a^{2}}}=\log|x+\sqrt{x^{2}-a^{2}}|+C\]

(v) \[\int\frac{dx}{\sqrt{a^{2}-x^{2}}}=\sin^{-1}\frac{x}{a}+C\]

(vi) \[\int\frac{dx}{\sqrt{x^{2}+a^{2}}}=\log|x+\sqrt{x^{2}+a^{2}}|+C\]

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`.

• 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 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 \(x^2-a^2\).

• 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 \[e^x \sin x\], \[e^x \cos x\].

• 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.

• 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.