What is the integration of log (1+x^2)?

Use integration by part to solve this question.

\[% \color{red}{\boxed{\color{blue}{\boxed{\color{black}{\text{ Using integration by parts, we have}\\\displaystyle \quad \int \ln \left(1+x^{2}\right) d x\\\displaystyle =x \ln \left(1+x^{2}\right)-\int x \frac{2 x}{1+x^{2}} d x \\\displaystyle =x \ln \left(1+x^{2}\right)-2 \int\left(1-\frac{1}{1+x^{2}}\right) d x \\\displaystyle =x \ln \left(1+x^{2}\right)-2 x+2 \tan ^{-1} x+C}}}}} \]

Comments

Area under the curve - Notes, Concept and All Important Formula

AREA UNDER THE CURVE 1. The area bounded by the curve \(y=f(x)\) , the \(x\) -axis and the ordinates \(x=a \) & \(x=b\) is given by, \(A=\displaystyle \int_{a}^{b} f(x) \,\, dx=\displaystyle \int_{a}^{b} y \,\, dx, f(x) \geq 0\) All Chapter Notes, Concept and Important Formula 2. If the area is below the \(x\) -axis then \(A\) is negative. The convention is to consider the magnitude only i.e. \(A=\left|\displaystyle \int_{a}^{b} y \,\, dx\right|\) in this case. 3. The area bounded by the curve \(x = f ( y ),\) y -axis \(\&\) abscissa \(y = c\) , \(y=d\) is given by, Area \(=\displaystyle \int_{c}^{d} x d y=\displaystyle \int_{c}^{d} f(y) d y, f(y) \geq 0\) 4. Area between the curves \(y=f(x) \) & \(y=g(x)\) between the ordinates \(x=a\) & \( x=b\) is given by, \(A=\displaystyle \int_{a}^{b} f(x) \,\, dx-\displaystyle \int_{a}^{b} g(x) \,\, dx\) \(=\displaystyle \int_{a}^{b}[f(x)-g(x)] \,\, dx, f(x) \geq g(x) \,\, \forall x \in(a, b)\) 5. Average...

What are Function and how its work on Calculus?

What are Function ? Table Of Contents Introduction with beautiful example Here's a plant, and what you see here is it's shadow. Can you list the things that the length of the shadow is dependent on. One, it's dependent on the position of the source of light. Anything else that you can think of. If the height of the plant grows then the shadows length will also change, right. So the length of the shadow is dependent on the position of the source of light, and the height of the plant too. So we can say that the length of the shadow is a function of the following two things. The output is dependent on these two things, which could be considered as the inputs. That's a very simple way to understand functions. Could you think of more inputs, this output is dependent on here, tell us yours answers in the comment section below. How do Function work in calculus ? That's what we'll see in this topics. Previously, we saw an idea to find the instant...

Relation (Mathematics) - Notes, Concept and All Important Formula

RELATIONS 1. INTRODUCTION : Let \(A\) and \(B\) be two sets. Then a relation \(R\) from \(A\) to \(B\) is a subset of \(A \times B\) . thus, \(R\) is a relation from \(A\) to \(B \Leftrightarrow R \subseteq A \times B\) . Total Number of Relations : Let \(A\) and \(B\) be two non-empty finite sets consisting of \(m\) and \(n\) elements respectively. Then \(A \times B\) consists of mn ordered pairs. So total number of subsets of \(A \times B\) is \(2^{m n}\) . Domain and Range of a relation : Let \(R\) be a relation from a set \(A\) to a set \(B\) . Then the set of all first components or coordinates of the ordered pairs belonging to \(R\) is called to domain of \(R\) , while the set of all second components or coordinates of the ordered pairs in \(R\) is called the range of \(R\) . Thus, \(\quad\) Domain \(( R )=\{ a :( a , b ) \in R \}\) and, Range \(( R )=\{ b :( a , b ) \in R \}\) It is evident from the definition that the domain of a relation from \(A\) to...

MATHEMATICAL WORLD

Search This Blog