Summary

This section gathers the key definitions, standard results and formulae of the Integration and Its Application unit of CBSE Class 12 Applied Mathematics (subject code 241) into a single quick-revision recap — useful for board-exam and previous-year-paper practice.

Integration and the indefinite integral

• Integration is the reverse process of differentiation. A function $F$ whose derivative is $f$ is called an anti-derivative (or primitive) of $f$.
• If $\dfrac{d}{dx}\big(F(x)+C\big)=f(x)$, then the anti-derivative of $f(x)$ is $F(x)+C$. Because the constant $C$ can take any value, this is called the indefinite integral, representing a whole family of curves.

Standard integral formulae

• $\displaystyle\int x^{n}\,dx=\frac{x^{n+1}}{n+1}+C,\quad n\neq-1$
• $\displaystyle\int 1\,dx=x+C$
• $\displaystyle\int e^{x}\,dx=e^{x}+C$
• $\displaystyle\int a^{x}\,dx=\frac{a^{x}}{\log a}+C$
• $\displaystyle\int \frac{1}{x}\,dx=\log|x|+C$
• $\displaystyle\int \frac{1}{\sqrt{x^{2}+a^{2}}}\,dx=\log\left(x+\sqrt{x^{2}+a^{2}}\right)+C$
• $\displaystyle\int \frac{1}{\sqrt{x^{2}-a^{2}}}\,dx=\log\left(x+\sqrt{x^{2}-a^{2}}\right)+C$
• $\displaystyle\int \sqrt{x^{2}+a^{2}}\,dx=\frac{x}{2}\sqrt{x^{2}+a^{2}}+\frac{a^{2}}{2}\log\left|x+\sqrt{x^{2}+a^{2}}\right|+C$
• $\displaystyle\int \sqrt{x^{2}-a^{2}}\,dx=\frac{x}{2}\sqrt{x^{2}-a^{2}}-\frac{a^{2}}{2}\log\left|x+\sqrt{x^{2}-a^{2}}\right|+C$
• $\displaystyle\int \frac{1}{x^{2}-a^{2}}\,dx=\frac{1}{2a}\log\left|\frac{x-a}{x+a}\right|+C$
• $\displaystyle\int \frac{1}{a^{2}-x^{2}}\,dx=\frac{1}{2a}\log\left|\frac{a+x}{a-x}\right|+C$

Integration by substitution and by partial fractions

• Substitution replaces the variable by a suitable function so that the integrand takes a standard form; the differential is transformed accordingly.
• A rational function $\dfrac{P(x)}{Q(x)}$ is proper when $\deg P<\deg Q$ and improper otherwise; an improper one is first reduced by long division. Proper rational functions are integrated by partial fractions, according to the type of the denominator:

Integration by parts

• Integration by parts: $\displaystyle\int f(x)\,g(x)\,dx=f(x)\int g(x)\,dx-\int\!\left[f'(x)\int g(x)\,dx\right]dx$. The first function $f(x)$ is chosen by the ILATE order (Inverse, Logarithmic, Algebraic, Trigonometric, Exponential).

The definite integral

• A definite integral is written $\displaystyle\int_{a}^{b} f(x)\,dx$, where $a$ is the lower limit and $b$ the upper limit; unlike an indefinite integral, it has a single fixed value.
• Second Fundamental Theorem of Calculus: if $f$ is continuous on $[a,b]$ and $F$ is an anti-derivative of $f$, then $\displaystyle\int_{a}^{b} f(x)\,dx=\big[F(x)\big]_{a}^{b}=F(b)-F(a)$.

Properties of the definite integral

• $\displaystyle\int_{a}^{b} f(x)\,dx=\int_{a}^{b} f(t)\,dt$
• $\displaystyle\int_{a}^{b} f(x)\,dx=-\int_{b}^{a} f(x)\,dx$
• $\displaystyle\int_{a}^{b} f(x)\,dx=\int_{a}^{c} f(x)\,dx+\int_{c}^{b} f(x)\,dx,\quad a<c<b$
• $\displaystyle\int_{a}^{b} f(x)\,dx=\int_{a}^{b} f(a+b-x)\,dx$
• $\displaystyle\int_{0}^{a} f(x)\,dx=\int_{0}^{a} f(a-x)\,dx$
• $\displaystyle\int_{0}^{2a} f(x)\,dx=\int_{0}^{a} f(x)\,dx+\int_{0}^{a} f(2a-x)\,dx$
• $\displaystyle\int_{0}^{2a} f(x)\,dx=2\int_{0}^{a} f(x)\,dx$ if $f(2a-x)=f(x)$, and $=0$ if $f(2a-x)=-f(x)$ …