Revision: Integral Calculas JEE Main Integral Calculas

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

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

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

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

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

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.

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

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.

• 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 \(x^2-a^2\).

• 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.

• 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, \(C\).

• 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.

• 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.

• A = ∫ (upper − lower) dx
• Find intersection points → solve f(x) = g(x)
• If the graph crosses → split into parts
• Final area = sum of all parts