Class 12th

Kindly consider the following Image 

Glycerol can be separated from spent-lye in soap industry by using reduce pressure distillation technique.

Students can find the formula of integration for a particular function in this article. It is important to memorise these formulas to solve the integral problem easily. Below is the integration of a particular function formula.

The formula of a particular function:

1. Power Rule: 

$\int x^{n}dx=\frac{x^{n+1}}{n+1}+C(n\ne -1)$

2. Reciprocal Function:

$\int \frac{1}{x}dx=ln\left|x\right|+C$

3. Trigonometric Function: 

• $\int sinxdx=-cosx+C$
• $\int cosxdx=sinx+C$
• $\int {sec}^{2}xdx=tanx+C$
• $\int {csc}^{2}xdx=-cotx+C$
• $\int secxtanxdx=secx+C$
• $\int cscxcotxdx=-cscx+C$

4. Inverse Trigonometric Function

• $\int \frac{1}{1+x^2}dx={tan}^{-1}x+C$
• $\int \frac{1}{\sqrt{1-x^2}}dx={sin}^{-1}x+C$
• $\int \frac{-1}{\sqrt{1-x^2}}dx={cos}^{-1}x+C$

? 2 adj (3 A adj (2A)|
= 23.? 3 A adj (2A)|
|2
= 23 ⋅ (33)2 ⋅ | A|2 ⋅ |adj (2 A)|2
= 23 ⋅ 36 ⋅ | A|2 ⋅ (|2 A|2)2
= 23 ⋅ 36 ⋅ | A|2 [ (23)2 ⋅ | A|2]2
= 23. 36. |A|2. 212. |A|4
= 215. 36. |A|6
= 215 ⋅ 36 ⋅ 56 = 2? ⋅ 3? ⋅ 5?
? ? = 15? = 6? = 6
? +? +? = 27

Kindly go through the solution

A = [ x 1 ]
[ 1 0 ]
A² = [ x 1 ] [ x 1 ] = [ x²+1 x ]
[ 1 0 ] [ 1 0 ] [ x 1 ]
A? = [ x²+1 x ] [ x²+1 x ]
[ x 1 ] [ x 1 ]
= [ (x²+1)²+x² x (x²+1)+x ]
[ x (x²+1)+x²+1 ]
a? = (x² + 1)² + x² = 109
⇒ x = ±3
a? = x² + 1 = 10

0 ≤ y ≤ x² + 1, 0 ≤ y ≤ x + 1, 1/2 ≤ x ≤ 2
Required area
= 19/24 + 5/2 = 79/24

2π - (sin? ¹ (4/5) + sin? ¹ (5/13) + sin? ¹ (16/65)
= 2π - (tan? ¹ (4/3) + tan? ¹ (5/12) + tan? ¹ (16/63)
= 2π - (tan? ¹ (63/16) + tan? ¹ (16/63)
= 2π - π/2 = 3π/2

f (x) = (3x - 7)x²/³
⇒ f (x) = 3x? /³ - 7x²/³
⇒ f' (x) = 5x²/³ - 14/ (3x¹/³)
= (15x - 14) / (3x¹/³) > 0

∴ f' (x) > 0 ∀x ∈ (-∞, 0) U (14/15, ∞)