The integration of 1/(1+x2) is equal to tan2 x. In this post, we will see how to integrate 1/(1+x^2). Integration of $\dfrac{1}{1+x^2}$ Question: What is the integration of $\dfrac{1}{1+x^2}$? That is, Find $\int \dfrac{1}{1+x^2} dx$ Answer: The integration of $\dfrac{1}{1+x^2}$ is $\tan^2 x$. Explanation: Let us substitute $x=\tan t$ $\cdots (\star)$ Differentiating with respect to … Read more
The integration of e3x is e3x/3. In this post, we will learn how to find the integral of e to the 3x. Let us recall the formula of the integral of emx: $\int e^{mx} dx=\dfrac{e^{mx}}{m}+C$ where C is an integral constant. Thus, the integral of e3x will be equal to $\int e^{3x} dx=\dfrac{e^{3x}}{3}+C$. What is the … Read more
The derivative of 1 by root x is -1/(2x root(x)). In this post, we will find the derivative of 1/root(x) by the first principle of derivatives. We will also find the integration of 1 by root x. Derivative of 1 by root x from First Principle The first principle of derivatives says that the derivative … Read more
The cube root of x is a number whose cube is x, in other words, (cube root of x)^3 = x. In this post, we will evaluate the integration of the cube root of x. To do so, we will use the power rule of derivatives which is given below: $\displaystyle \int x^n dx = … Read more
The integration of root x is equal to 2/3 x3/2+C. In this post, we will calculate the integration of the square root of x. Note that the square root of x is an algebraic function. We will use the following power rule of derivatives: $\int x^n dx = \dfrac{x^{n+1}}{n+1}+c$ $ \cdots (i)$ where $c$ is … Read more
The integration of log(sin x) from 0 to pi/2 is -π/2 log2. In this section, we will find the integral of log sinx from 0 to pi/2. The following property will be used to calculate the integral. $\int_0^af(x)dx=\int_0^af(a-x)dx$ …(i) Integral of log(sin x) from 0 to pi/2 Question: Show that $\int_0^{\frac{\pi}{2}} \log \sin x dx … Read more
Introduction to Integral Calculus: In Differential Calculus, we have learned how to find the derivative/differential of a differentiable function. The study of the inverse method of differential calculus is the main purpose of integral Calculus. We now understand this with an example. Let f(x) be a function with $f'(x)=\frac{d}{dx}$(f(x)) = 3×2. From this information, can … Read more
Integration by Substitution Method: Some Important Forms with Examples Let f(x) be an integrable function in one variable x. If we have to integrate some well-known twisted form of f(x), then we may take the help of the substitution method. Here we list some important forms. Form 1: $\int f(ax+b) dx$ can be solved by the … Read more
The following topics of Integral Calculus will be covered. Please click on the link provided below you wish to read. An Introduction to Integral Calculus Integration: Definition, Properties and Formulas Proofs of all integration formulas (to be updated) Integration by substitution Integration by parts (to be updated) Integration of some special functions (to be updated) Properties of definite … Read more
List of all Integral formulas | Special Integration Formula | Integration formulas with examples —————————————————————- Some Basic Integration Formulas 1. $\int dx=x+c$ 2. $\int x^n dx=\frac{x^{n+1}}{n+1}+c; (n \neq -1)$ 3. $\int e^x dx=e^x+c$ 4. $\int a^x dx=\frac{a^x}{\log|a|}+c; (a>0$ but $a \neq 1)$ 5. $\int \frac{1}{x} dx=\log x+c$ 6. $\int \frac{dx}{\sqrt{1-x^2}}=\sin^{-1} x+c$ 7. $\int \frac{dx}{1+x^2}=\tan^{-1} … Read more