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Category: Integration

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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( […]

Integrate limit 0 to two pie (1 divided by 1 plus e to the power sin x) dx

Evaluate :   Let    Adding equations (1) and (2)

INTEGRATION AS INVERSE PROCESS OF DIFFERENTIATION

Integral means anti – derivative of a function. If , then is anti – derivative of or . will have infinite number of values, so it is called indefinite integral of . {Derivative of any constant is zero} . where c is any constant called arbitrary or integration constant. We must write +c in every […]