Physics > Basic Mathematics and Measurements > 5.0 Integration
Basic Mathematics and Measurements
1.0 Introduction
2.0 Trigonometry
2.1 Values of trigonometric angles
2.2 Trigonometric identities
2.3 Trigonometric functions in different quadrants
3.0 Basic logarithmic functions
4.0 Differentiation
4.1 Derivatives of some simple functions
4.2 Rules of differentiation
4.3 Application of differentiation
4.4 Solved examples of differentiation
5.0 Integration
6.0 Graphs
6.1 Straight line
6.2 Circle
6.3 Ellipse
6.4 Parabola
6.5 Rectangular hyperbola
6.6 Exponential function
6.7 Logarithmic functions
7.0 Significant Figures
7.1 Rules to determine the significant figures
7.2 Rules for arthimetic operation with significant figures
8.0 Rounding off
9.0 Errors
9.1 Systematic error
9.2 Random errors
9.3 Least count error
9.4 Absolute error
9.5 Mean absolute error
9.6 Relative error or fractional error
9.7 Percentage error
10.0 Combination of errors
10.1 Addition of errors
10.2 Subtraction of errors
10.3 Multiplication of errors
10.4 Division of errors
10.5 Power
11.0 Length Measuring Instruments
11.1 Vernier Callipers
11.2 Zero error of vernier calliper
11.3 Vernier calliper solved examples
11.4 Screw Gauge
11.5 Zero error of screw gauge
11.6 Screw gauge solved examples
12.0 Questions
5.2 Rules of integration
2.2 Trigonometric identities
2.3 Trigonometric functions in different quadrants
4.2 Rules of differentiation
4.3 Application of differentiation
4.4 Solved examples of differentiation
6.2 Circle
6.3 Ellipse
6.4 Parabola
6.5 Rectangular hyperbola
6.6 Exponential function
6.7 Logarithmic functions
7.2 Rules for arthimetic operation with significant figures
9.2 Random errors
9.3 Least count error
9.4 Absolute error
9.5 Mean absolute error
9.6 Relative error or fractional error
9.7 Percentage error
10.2 Subtraction of errors
10.3 Multiplication of errors
10.4 Division of errors
10.5 Power
11.2 Zero error of vernier calliper
11.3 Vernier calliper solved examples
11.4 Screw Gauge
11.5 Zero error of screw gauge
11.6 Screw gauge solved examples
(A). Multiplication by constant: $\int {cf(x)dx} = c\int {f(x)dx} $
(B). Sum rule: $ \int {\left[ {f(x) + g(x)} \right]} dx = \int {f(x)dx} + \int {g(x)dx} $
(C). Difference rule: $\int {\left[ {f(x) - g(x)} \right]} dx = \int {f(x)dx} - \int {g(x)dx} $
(D). Integration by parts: $\int {uvdx} = u\int {vdx} - \int {\frac{{du}}{{dx}}\left( {\int {vdx} } \right)} \,dx$
(E). Integration by substitution: $\int {f\left( {g(x)} \right)} g'(x)dx = \int {f(u)du} $
where $g\left( x \right) = u$
Proof: Given, $$I = \int {f\left( {g(x)} \right)} g'(x)dx$$
Let, $$g\left( x \right) = u$$
Differentiating both sides we get,
$$g'(x) = \frac{{du}}{{dx}}$$$$du = g'(x)dx$$
Substituting the value in the equation we get,
$$I = \int {f(u)du} $$
