Integration as an Inverse Process of Differentiation - Practice Questions & MCQ

Integration is the reverse process of differentiation. In integration, we find the function whose differential coefficient is given. 

For example,

$\begin{array}{rl}& \frac{d}{dx}(\sin x)=\cos x\\ & \frac{d}{dx}\left(x^2\right)=2x\\ & \frac{d}{dx}\left(e^x\right)=e^x\end{array}$

In the above example, the function cos(x) is the derivative of sin(x). We say that sin(x) is an anti-derivative (or an integral) of cos(x). Similarly, x² and e^x are the antiderivatives (or integrals) of 2x and e^x respectively.  

Also note that the derivative of a constant  (C) is zero. So we can write the above examples as:

$\begin{array}{rl}& \frac{d}{dx}(\sin x+c)=\cos x\\ & \frac{d}{dx}(x^2+c)=2x\\ & \frac{d}{dx}(e^x+c)=e^x\end{array}$

Thus, the anti-derivatives (or integrals) of the above functions are not unique. Actually, there exist infinitely many anti-derivatives of each of these functions which can be obtained by selecting C arbitrarily from the set of real numbers. 

For this reason, C is referred to as an arbitrary constant. In fact, C is the parameter by varying which one gets different anti-derivatives (or integrals) of the given function. 

If the function F(x) is an antiderivative of f(x), then the expression F(x) + C is the indefinite integral of the function f(x) and is denoted by the symbol ∫ f(x) dx. 

By definition,

$\int \mathrm{f}(\mathrm{x})\mathrm{dx}=\mathrm{F}(\mathrm{x})+\mathrm{c},$ where ${\mathrm{F}}^{\prime }(\mathrm{x})=\mathrm{f}(\mathrm{x})$ and ' ${\mathrm{c}}^{\prime }$ is constant.

Here, 

Rules of integration

f(x) and g(x) are functions with antiderivatives ∫ f(x) and  ∫ g(x) dx. Then,

(a) $\int \mathrm{kf}(x)dx=k\int f(x)dx$ for any constant $k$.
(b) $\int (f(x)+g(x))dx=\int f(x)dx+\int g(x)dx$
(c) $\int (f(x)-g(x))dx=\int f(x)dx-\int g(x)dx$

Standard Integration Formulae

Since,$\frac{\mathrm{d}}{\mathrm{dx}}(\mathrm{F}(\mathrm{x}))=\mathrm{f}(\mathrm{x})\iff \int \mathrm{f}(\mathrm{x})\mathrm{dx}=\mathrm{F}(\mathrm{x})+\mathrm{c}$

Based on this definition and various standard formulas (which we studied in Limit, Continuity, and Differentiability) we can obtain the following important integration formulae, 

1.$\frac{\mathrm{d}}{\mathrm{dx}}(\mathrm{kx})=\mathrm{k}\Rightarrow \int \mathrm{kdx}=\mathrm{kx}+\mathrm{C}$, where k is a constant
2.$\frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{{\mathrm{x}}^{\mathrm{n}+1}}{\mathrm{n}+1}\right)={\mathrm{x}}^{\mathrm{n}},\mathrm{n}\ne -1\Rightarrow \int {\mathrm{x}}^{\mathrm{n}}\mathrm{dx}=\frac{{\mathrm{x}}^{\mathrm{n}+1}}{\mathrm{n}+1}+\mathrm{C},\mathrm{n}\ne -1$
3. $\frac{\mathrm{d}}{\mathrm{dx}}(\log |\mathrm{x}|)=\frac{1}{\mathrm{x}}\Rightarrow \int \frac{1}{\mathrm{x}}\mathrm{dx}=\log |\mathrm{x}|+\mathrm{C}$, when $\mathrm{x}\ne 0$
4. $\frac{\mathrm{d}}{\mathrm{dx}}\left({\mathrm{e}}^{\mathrm{x}}\right)={\mathrm{e}}^{\mathrm{x}}\Rightarrow \int {\mathrm{e}}^{\mathrm{x}}\mathrm{dx}={\mathrm{e}}^{\mathrm{x}}+\mathrm{C}$
5.$\frac{\mathrm{d}}{\mathrm{dx}}\left(\frac{{\mathrm{a}}^{\mathrm{x}}}{{\log }_{\mathrm{e}}\mathrm{a}}\right)={\mathrm{a}}^{\mathrm{x}},\mathrm{a}>0,\mathrm{a}\ne 1\Rightarrow \int {\mathrm{a}}^{\mathrm{x}}\mathrm{dx}=\frac{{\mathrm{a}}^{\mathrm{x}}}{{\log }_{\mathrm{e}}\mathrm{a}}+\mathrm{C}$