Revision: Integration Maths HSC Commerce (English Medium) 12th Standard Board Exam Maharashtra State Board

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

• \[\int\frac{1}{x^2-a^2}dx=\frac{1}{2a}log\left|\frac{x-a}{x+a}\right|+c\]

• \[\int\frac{1}{a^2-x^2}dx=\frac{1}{2a}log\left|\frac{a+x}{a-x}\right|+c\]

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

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

• \[\int
  \begin{bmatrix}
  f(x)
  \end{bmatrix}^nf^{\prime}(x)dx=\frac{
  \begin{bmatrix}
  f(x)
  \end{bmatrix}^{n+1}}{(n+1)}+c\]

• \[\int\left[\frac{f^{\prime}(x)}{f(x)}\right]dx=\log f(x)+c\]

• \[\int\left[\frac{f^{\prime}(x)}{\sqrt{f(x)}}\right]dx=2\sqrt{f(x)}+c\]

• \[\int\left[\frac{f^{\prime}(x)}{\sqrt[n]{f(x)}}\right]dx=\frac{n\sqrt[n]{\left[f(x)\right]^{n-1}}}{n-1}+c\]

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 x = ϕ(t), then \[\int f(x)dx=\int f(\phi(t))\phi^{\prime}(t)dt\]

If u and v are two functions of x, then

\[\int u.vdx=u\int vdx-\int\left[\int vdx.\frac{du}{dx}\right]dx\]

• 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.

Sum Rule:

\[\int[f(x)+g(x)]dx=\int f(x)dx+\int g(x)dx\]

Difference Rule:

\[\int[f(x)-g(x)]dx=\int f(x)dx-\int g(x)dx\]

Constant Multiple Rule:

\[\int kf(x)dx=k\int f(x)dx\]

Substitution Result:

If  \[\int f(x)dx=F(x)+C\] Then \[\int f(ax+b)dx=\frac{1}{a}F(ax+b)+C\]

L → A → E

• L = Logarithmic (log x)

• A = Algebraic (x, x², polynomial)

• E = Exponential (eˣ)