Integration Rules: Overview, Questions, Preparation

Integration is the act of bringing together smaller components into a single system that functions as one. To numerically apply this operation, some rules must be followed. 

Rules of integration

1. Integral of a power function: 

While integrating a function with a degree greater than 1, the following rule is applied. 

∫xⁿ dx = x^{n + 1} / (n + 1) + c

2. Integral of a function multiplied with a constant k:

To integrate a function with a coefficient, this rule is applied. 

∫k f(x) dx = k ∫f(x) dx

3. Integral of a sum or difference of functions 

To integrate a sum of functions, the following rule is applied. 

 ∫[f(x) + g(x)] dx = ∫f(x) dx + ∫g(x) dx

To integrate a difference of functions, the following rule is applied. 

∫[f(x) - g(x)] dx = ∫f(x) dx - ∫g(x) dx

4. Integral a product of two functions.

This rule is also called integration by parts. 

∫u v dx = u∫v dx −∫u' (∫v dx) dx

5. Integration by substitution 

This rule is applied when the equation to be integrated contains a function and the differential of the function in some form. The function will be substituted by some variable ( say ‘x’) and the derivative of the function will be substituted by dx. This simplifies the integration process. 

Importance and Weightage

Students are familiarised with integration in class 12. Having a command over these rules will assist students in figuring out the right approach of solving complex numericals. They will be continuously exposed to integration in different areas of Mathematics and Science.

1. (2x² +e^x).dx

Solution. Using ∫[f(x) + g(x)] dx = ∫f(x) dx + ∫g(x) dx,

∫(2x² +e^x) dx= ∫2x² dx + e^x dx.

Using ∫k f(x) dx = k ∫f(x) dx

∫(2x² +e^x) dx=2∫x² dx + ∫e^x dx.

Using ∫xⁿ dx = x^{n + 1} / (n + 1) + c,

∫(2x² +e^x) dx=  ⅔ x³+ e^x+ C

2. ∫{(x³+ 5x²- 4)/x²}.dx

Solution. ∫{(x³+ 5x²- 4)/x²}.dx = (x +  5 - 4 x^-2) dx

Using  ∫[f(x) + g(x)] dx = ∫f(x) dx + ∫g(x) dx

= ∫x dx + ∫5 dx - ∫4x^-2 dx

 Using ∫k f(x) dx = k ∫f(x) dx and ∫x ⁿ dx = x ^{n + 1} / (n + 1) + c,

=  x²/ 2 + 5∫dx - 4∫x^-2 dx

Using ∫xⁿ dx = x^{n + 1} / (n + 1) + c

=  x²/ 2 + 5x + 4/x + C

3. (ax2+ bx + c) dx

Solution. Using,  ∫[f(x) + g(x)] dx = ∫f(x) dx + ∫g(x) dx

(ax²+ bx + c) dx = ∫(ax²) dx+ ∫bx dx + ∫c dx

Using, ∫k f(x) dx = k ∫f(x) dx

= a∫x² dx+ b∫x dx + c∫ dx

Using, ∫xⁿ dx = x^{n + 1} / (n + 1) + c

=( ax³)/ 3 dx+ (b x² )/2  + x

4. (log x)²/x

Solution. We will use the substitution rule to solve this problem. 

log x = t

(1/x)dx = dt

∫(log x)²/x  dx = dt

= t³/ 3 + C

= (log x)³/ 3 + C 

Q: What is dx?

Q: What is the significance of learning integrals?

Q: Are trigonometric functions f(x) functions?

Q: Can we perform elemental operations with integration problem solutions?

Q: Can there be an integration within an integration?