Revision: Definite Integration Maths and Stats HSC Science (General) 12th Standard Board Exam Maharashtra State Board

The representation \[\int_{x=a}^{x=b}\] f(x)dx is called the definite integral of f(x) from x = a to x = b.

F(x) = ∫f(x) dx is called the indefinite (without any limits on x) integral of f(x).

• Definite integral = limit of Riemann sum

• It represents the area under the curve from x = a to x = b

• $\backslash [\backslash int\_a^bf(x)dx=\backslash lim\_\{n\backslash to\backslash infty\}\backslash sum\_\{r=0\}^\{n-1\}f(t\_r)(x\_\{r+1\}-x\_r)\backslash ]$
• $\backslash [g\backslash left(b\backslash right)-g\backslash left(a\backslash right)=\backslash lim\_\{n\backslash to\backslash infty\}\backslash sum\_\{a\}\backslash left(x\_\{r+1\}-x\_\{r\}\backslash right)\backslash cdot\; f\backslash left(t\_\{r\}\backslash right)=\backslash lim\_\{n\backslash to\backslash infty\}S\_\{n\}=\backslash int\_\{a\}^\{b\}f\backslash left(x\backslash right)dx\backslash ]$

\[\begin{aligned}
\int_{0}^{\pi/2}\sin^{n}xdx & =\quad\frac{(n-1)}{n}\cdot\frac{(n-3)}{(n-2)}\cdot\frac{(n-5)}{(n-4)}\cdot\cdot\cdot\frac{4}{5}\frac{2}{3},\quad\mathrm{if~}n\mathrm{~is~odd}. \\
 & =\quad\frac{(n-1)}{n}\cdot\frac{(n-3)}{(n-2)}\cdot\frac{(n-5)}{(n-4)}\cdot\cdot\cdot\frac{3}{4}\frac{1}{2}\cdot\frac{\pi}{2},\quad\mathrm{if~}n\mathrm{~is~even}.
\end{aligned}\]

\[\int_{0}^{\pi_{2}}\cos^{n}xdx=\int_{0}^{\pi_{2}}\left[\cos\left(\frac{\pi}{2}-0\right)\right]^{n}dx=\int_{0}^{\pi_{2}}\left[\sin x\right]^{n}dx=\int_{0}^{\pi_{2}}\sin^{n}xdx\]

Let f be the continuous function defined on [a, b] and if \[\int f(x)dx=g(x)+c\] then \[\int_{a}^{b}f\left(x\right)dx=g\left(b\right)-g\left(a\right)\]

Property I:

\[\int_{a}^{a}f(x)dx=0\]

Property II:

\[\int_{a}^{b}f\left(x\right)dx=-\int_{b}^{a}f\left(x\right)dx\]

Property III:

\[\int_{a}^{b}f(x)dx=\int_{a}^{b}f(t)dt\] 

Property IV:

\[\int_{a}^{b}f\left(x\right)dx=\int_{a}^{c}f\left(x\right)dx+\int_{c}^{b}f\left(x\right)dx\]

where a < c < b i.e. c ∈ [a,    b] 

Property V:

\[\int_{a}^{b}f(x)dx=\int_{a}^{b}f(a+b-x)dx\]

Property VI:

\[\int_{0}^{a}f(x)dx=\int_{0}^{a}f(a-x)dx\]

Property VII:

\[\int_{0}^{2a}f\left(x\right)dx=\int_{0}^{a}f\left(x\right)dx+\int_{0}^{a}f\left(2a-x\right)dx\]

Property VIII:

$$\int_{-a}^{a} f(x) \, dx = 2 \cdot \int_{0}^{a} f(x) \, dx \quad , \text{ if } f(x) \text{ even function}$$
$$
= 0 \quad , \text{ if } f(x) \text{ is odd function}$$