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### Calculus - Integral Calculus Problem Set V

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## Calculus - Integral Calculus Problem Set V - Outline of Contents:

### In this tutorial, we'll pick up a variety of examples and solved problems - More complex examples of definite and indefinite integration, examples of integration as the limit of a summation of a series.

Here's a quick look at the kind of integrals you can expect to encounter in this tutorial :

(4ex + 6e-x)
∫    --------------    dx                       we'll show you how to solve this by making the substitution t = e
9ex − 4e-x

∫ sech x dx                                       we'll show you how to solve this by making the substitution t = e
0

π/2
∫ dx / (5 + 7 cos x + sin x)                we'll show you how to solve this by making the substitution t = tan (x/2)
-π/2

∫ dx / (a2 sin2 x + b2 cos2 x)              we'll show you how to solve this by making the substitution t =tan x

∫ sin x cos x dx / (a2 cos2 x + b2 sin2 x)                 we solve this by making the substitution t =tan2 x

π/2
∫ sin x  dx / (sin x + cos x)                           We use properties of definite integrals to tackle cases like this.
0

π/2
∫ sin2 x  dx / (sin x + cos x)  =  (log (1 + √ 2))  / √ 2
0

b
ex dx as the limit of a sum.
a

b
∫ x2 dx as the limit of a sum.
a

Find the limit, when n tends to inﬁnity, of the series
n2 / (n2 + 1) 2  + n2 / (n2 + 222  +  n2 / (n2 + 32 .....  n2 / (n2 + (n-1)22

We will try to solve these integrals, and a lot more interesting cases as well, which involve partial fractions and some interesting substitutions.

### In Case you'd like to take a look at some of our other tutorials related to Single Variable Calculus :

 Quick and introductory definitions related to Funtions, Limits and Continuity - Defining the domain and range of a function, the meaning of continuity, limits, left and right hand limits, properties of limits and the "lim" operator; some common limits;  defining the L'Hospital rule, intermediate and extreme value theorems. Functions, Limits and Continuity - Solved Problem Set I - The Domain, Range, Plots and Graphs of Functions;  L'Hospital's Rule- -  Solved problems demonstrating how to compute the domain and range of functions, drawing the graphs of functions, the mod function, deciding if a function is invertible or not; calculating limits for some elementary examples, solving 0/0 forms, applying L'Hospital rule. Functions, Limits and Continuity - Solved Problem Set II - Conditions for Continuity, More Limits, Approximations for ln (1+x) and sin x for infinitesimal values of x   - More advanced cases of evaluating limits, conditions for continuity of functions, common approximations used while evaluating limits for ln ( 1 + x ), sin (x); continuity related problems for more advanced functions than the ones in the first group of problems (in the last tutorial). Functions, Limits and Continuity - Solved Problem Set III - Continuity and Intermediate Value Theorems - Problems related to Continuity, intermediate value theorem. Introductory concepts and definitions related to Differentiation - Basic formulas, Successive Differentiation, Leibnitz, Rolle and Lagrange Theorems, Maxima , Minima, Convexity, Concavity, etc - Theory and definitions introducing differentiability, basic differentiation formulas of common algebraic and trigonometric functions , successive differentiation, Leibnitz Theorem, Rolle's Theorem,  Lagrange's Mean Value Theorem, Increasing and decreasing functions, Maxima and Minima; Concavity, convexity and inflexion, implicit differentiation. Differential Calculus - Solved Problem Set I - Common Exponential, Log , trigonometric and polynomial functions - Examples and solved problems - differentiation of common algebraic, exponential, logarithmic, trigonometric and polynomial functions and terms; problems related to differentiability . Differential Calculus - Solved Problem Set II - Derivability and continuity of functins - Change of Indepndent Variables - Finding N-th Derivatives - Examples and solved problems - related to derivability and continuity of functions; changing the independent variable in a differential equation; finding the N-th derivative of functions Differential Calculus - Solved Problems Set III- Maximia, Minima, Extreme Values,  Rolle's Theorem - Examples and solved problems - related to increasing and decreasing functions; maxima, minima and extreme values; Rolle's Theorem Differential Calculus - Solved Problems Set IV - Points of Inflexion, Radius of Curvature, Curve Sketching -  Examples and solved problems - Slope of tangents to a curve, points of inflexion, convexity and concavity of curves, radius of curvature and asymptotes of curves, sketching curves Differential Calculus - Solved Problems Set V - Curve Sketching, Parametric Curves - More examples of investigating and sketching curves, parametric representation of curves Introducing Integral Calculus - Definite and Indefinite Integrals - using Substitution , Integration By Parts, ILATE rule  - Theory and definitions. What integration means, the integral and the integrand. Indefinite integrals, integrals of common functions.  Definite integration and properties of definite integrals; Integration by  substitution, integration by parts, the LIATE rule, Integral as the limit of a sum. Important forms encountered in integration. Integral Calculus - Solved Problems Set I - Basic examples of polynomials and trigonometric functions, area under curves - Examples and solved problems - elementary examples of integration involving trigonometric functions, polynomials; integration by parts; area under curves. Integral Calculus - Solved Problems Set II - More integrals, functions involving trigonometric and inverse trigonometric ratios - Examples and solved problems - integration by substitution, definite integrals, integration involving trigonometric and inverse trigonometric ratios. Integral Calculus - Solved Problems Set III - Reduction Formulas, Using Partial FractionsI- Examples and solved problems - Reduction formulas, reducing the integrand to partial fractions, more of definite integrals Integral Calculus - Solved Problems Set IV - More of integration using partial fractions, more complex substitutions and transformations - Examples and solved problems - More of integrals involving partial fractions, more complex substitutions and transformations Integral Calculus - Solved Problems Set V- Integration as a summation of a series - Examples and solved problems - More complex examples of integration, examples of integration as the limit of a summation of a series Introduction to Differential Equations and Solved Problems - Set I - Order and Degree, Linear and Non-Linear Differential Equations, Homogeneous Equations, Integrating Factor -  Theory and definitions. What a differential equation is; ordinary and partial differential equations; order and degree of a differential equation; linear and non linear differential equations; General, particular and singular solutions; Initial and boundary value problems; Linear independence and dependence; Homogeneous equations; First order differential equations; Characteristic and auxiliary equations. Introductory problems demonstrating these concepts. Introducing the concept of Integrating Factor (IF). Differential Equations - Solved Problems - Set II - D operator, auxillary equation, General Solution - Examples and solved problems - Solving linear differential equations, the D operator, auxiliary equations. Finding the general solution ( CF + PI ) Differential Equations - Solved Problems - Set III - More Differential Equations - More complex cases of differential equations. Differential Equations - Solved Problems - Set IV - Still more differential equations.