Maths

The integration of an odd function over a symmetric interval can be understood through some fundamental concepts in calculus and symmetry. An odd function is defined as a function $f(x)$ that satisfies the condition $f(-x) = -f(x)$ for all $x$ in the domain of the function. Graphically, odd functions exhibit symmetry about the origin, meaning

Definite Integrals if  $\int f(x)  dx= g(x)$$\int_{a}^{b} f(x) dx =g(b) -g(a)$ Properties of Definite Integrals (I) $\int_{a}^{b} f(x) dx= \int_{a}^{b} f(t) dt $ (II) $\int_{a}^{b} f(x) dx=- \int_{b}^{a} f(x) dx $ (III) $\int_{a}^{b} f(x) dx= \int_{a}^{c} f(x) dx + \int_{c}^{b} f(x) dx  $ where a<c <b (IV) $\int_{a}^{b} f(x) dx=\int_{a}^{b} f((a+b-x)) dx$ (V) $\int_{0}^{a} f(x)

Integration of sin square x can be calculated using trigonometric identities .Here is the formula for it $$ \int \sin^2(x) \, dx = \frac{1}{2} x – \frac{1}{4} \sin(2x) + C $$ Integration of sin 2x can be calculated using integration by substitutions .Here is the formula for it $$ \int \sin(2x) \, dx= \frac {-cos

The integration of cot square x $\cot ^2 x$, can be found using trigonometry identities and fundamental integral formulas . The integral of $\cot^2 x$ with respect to (x) is: \[\int \cot^2 x \, dx =-\cot x -x + C\] Here, (C) represents the constant of integration, which is added because the process of integration

The integration of tan square x $\tan ^2 x$, can be found using trigonometry identities and fundamental integral formulas . The integral of $\tan ^2 x$ with respect to (x) is: \[\int \tan^2 x \, dx =\tan x -x + C\] Here, (C) represents the constant of integration, which is added because the process of

integration of inverse trigonometric functions can be calculated using integration by parts ,integration by substitution .Here are the formula for it \[ \int \sin^{-1}x \, dx = x \sin^{-1}x + \sqrt{1 – x^2} + C \] \[ \int \cos^{-1}x \, dx = x \cos^{-1}x – \sqrt{1 – x^2} + C \] \[ \int \tan^{-1}x \,