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## Category: Mathematics

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### Integrate root cos2x divided by sinx with respect to x

$$\displaystyle \int{\frac{\sqrt{\cos 2x}}{\sin x}dx}$$ $$\displaystyle =\int{\frac{\sqrt{\frac{1-{{\tan }^{2}}x}{1+{{\tan }^{2}}x}}}{\sin x}}\,\,dx$$ $$\displaystyle =\int{\frac{\sqrt{1-{{\tan }^{2}}x}}{\sec x.\sin x}\,dx}$$ $$\displaystyle =\int{\frac{\sqrt{1-{{\tan }^{2}}x}}{\tan x}\,\,dx}$$ $$\displaystyle =\int{\frac{\sqrt{1-{{\tan }^{2}}x}}{\tan x\left( 1+{{\tan }^{2}}x \right)}\,{{\sec }^{2}}x\,\,dx}$$ $$\displaystyle =\int{\frac{\sqrt{1-{{\tan }^{2}}x}}{{{\tan }^{2}}x\left( 1+{{\tan }^{2}}x \right)}}\,\,\tan x{{\sec }^{2}}x\,\,dx$$ Let $$\displaystyle \sqrt{1-{{\tan }^{2}}x}\,\,=t$$ $$\displaystyle 1-{{\tan }^{2}}x={{t}^{2}}$$ Differentiate both sides w.r.t x $$\displaystyle -2\tan x{{\sec }^{2}}x\,\,dx\,\,=2\,tdt$$ \(\displaystyle \Rightarrow I=-\int{\frac{t}{\left( 1-{{t}^{2}} \right)\left( […]

### MULTIPLICATION TECHNIQUES

Let us consider two digit numbers ab and cd. For example: 23 × 45 Now, Now, we can try in one step.

### TRANSFORMATION FORMULAE

Formula to transform the product into sum or difference The sine formula of the sum and difference of two angles is written as follows: Add both the expression as follows: Then, subtract expression (2) from expression (1) as follows: Similarly consider the cosine formula of the sum and difference of two angles is written as […]

### PROPERTIES OF ARITHMETIC PROGRESSIONS

1. If each term of an A.P are added or subtracted with the same constant, then the resulting sequence is also in A.P with the same common difference. Example: 2, 5, 8, 11, 14, …………………….. . are in A.P. d = 3 when k = 2 is added in each term 4, 7, 10, 13, […]