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

Ellipse - Notes, Concept and All Important Formula

ELLIPSE 1. STANDARD EQUATION & DEFINITION : Standard equation of an ellipse referred to its principal axis along the co-ordinate axis is \(\dfrac{\mathbf{x}^{2}}{\mathbf{a}^{2}}+\dfrac{\mathbf{y}^{2}}{\mathbf{b}^{2}}=\mathbf{1}\) . where \(a>b \) & \( b^{2}=a^{2}\left(1-e^{2}\right)\) \(\Rightarrow a^{2}-b^{2}=a^{2} e^{2} .\) where \(e=\) eccentricity \((0<e<1)\) . \(\mathrm{FOCI}: \mathrm{S} \equiv(\mathrm{ae}, 0) \) & \( \mathrm{~S}^{\prime} \equiv(-\mathrm{ae}, 0) .\) (a) Equation of directrices : \(\mathrm{x}=\dfrac{\mathrm{a}}{\mathrm{e}} \) & \( \mathrm{x}=-\dfrac{\mathrm{a}}{\mathrm{e}} \text { . }\) (b) Vertices: \(\mathrm{A}^{\prime} \equiv(-\mathrm{a}, 0) \quad \) & \( \mathrm{~A} \equiv(\mathrm{a}, 0)\) (c) Major axis : The line segment \(A^{\prime} A\) in which the foci \(S^{\prime}\) & S lie is of length \(2 \mathrm{a} \) & \(\) is called the major axis \((a>b)\) of the ellipse. Point of intersection of major axis with dir...

Matrices - Notes, Concept and All Important Formula

MATRICES 1. INTRODUCTION: A rectangular array of mn numbers in the form of \(m\) horizontal lines (called rows) and \(n\) vertical lines (called columns), is called a matrix of order \(m\) by \(n\) , written as \(m \times n\) matrix. In compact form, the matrix is represented by \(\mathrm{A}=\left[\mathrm{a}_{\mathrm{i}}\right]_{\mathrm{mx}}\) All Chapter Notes, Concept and Important Formula 2. SPECIAL TYPE OF MATRICES : (a) Row Matrix (Row vector) : \(A=\left[a_{11}, a_{12}, \ldots \ldots \ldots . a_{1 n}\right]\) i.e. row matrix has exactly one row. (b) Column Matrix (Column vector) : \(A=\left[\begin{array}{c}a_{11} \\ a_{21} \\ : \\ a_{m 1}\end{array}\right]\) i.e. columr matrix has exactly one column. (c) Zero or Null Matrix : \(\left(A=O_{m \times n}\right)\) , An \(m \times n\) matrix whose all entries are zero. (d) Horizontal Matrix : A matrix of order \(\mathrm{m} \times \mathrm{n}\) is a horizontal matrix if \(\mathrm{n}>\mathrm{m}\) . (e) Vertical Matrix : ...

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

FUNCTION 1. DEFINITION : If to every value (considered as real unless other-wise stated) of a variable \(\mathrm{x}\) , which belongs to a set \(\mathrm{A}\) , there corresponds one and only one finite value of the quantity \(y\) which belong to set \(B\) , then \(y\) is said to be a function of \(x\) and written as \(f: A \rightarrow B, y=f(x), x\) is called argument or independent variable and \(y\) is called dependent variable. Pictorially: \( \underset{\text { input }}{\stackrel{\mathrm{x}}{\longrightarrow}}\boxed{\mathrm{f}}\,\, \underset{\text { output }}{\stackrel{\mathrm{f(x)=y}}{\longrightarrow}}\) \(y\) is called the image of \(x\) and \(x\) is the pre-image of \(y\) , under mapping \(\mathrm{f}\) . Every function \(\mathrm{f}: \mathrm{A} \rightarrow \mathrm{B}\) satisfies the following conditions. (i) \(f \subset A \times B\) (ii) \(\forall \,\, a \in A \,\, \exists\,\, b \in B\) such that \((a, b) \in f\) and (iii) If \((a, b) \in f \) &...

MATHEMATICAL WORLD

Search This Blog