Skip to main content

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

Use integration by part to solve this question.

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

Labels:

Comments

Post a Comment

Popular posts from this blog

Indefinite Integration - Notes, Concept and All Important Formula

INDEFINITE INTEGRATION If  f & F are function of \(x\) such that \(F^{\prime}(x)\) \(=f(x)\) then the function \(F\) is called a PRIMITIVE OR ANTIDERIVATIVE OR INTEGRAL of \(\mathrm{f}(\mathrm{x})\) w.r.t. \(\mathrm{x}\) and is written symbolically as \(\displaystyle \int \mathrm{f}(\mathrm{x}) \mathrm{d} \mathrm{x}\) \(=\mathrm{F}(\mathrm{x})+\mathrm{c} \Leftrightarrow \dfrac{\mathrm{d}}{\mathrm{dx}}\{\mathrm{F}(\mathrm{x})+\mathrm{c}\}\) \(=\mathrm{f}(\mathrm{x})\) , where \(\mathrm{c}\) is called the constant of integration. Note : If \(\displaystyle \int f(x) d x\) \(=F(x)+c\) , then \(\displaystyle \int f(a x+b) d x\) \(=\dfrac{F(a x+b)}{a}+c, a \neq 0\) All Chapter Notes, Concept and Important Formula 1. STANDARD RESULTS : (i) \( \displaystyle \int(a x+b)^{n} d x\) \(=\dfrac{(a x+b)^{n+1}}{a(n+1)}+c ; n \neq-1\) (ii) \(\displaystyle \int \dfrac{d x}{a x+b}\) \(=\dfrac{1}{a} \ln|a x+b|+c\) (iii) \(\displaystyle \int e^{\mathrm{ax}+b} \mathrm{dx}\) \(=\dfrac{1}{...
Post a Comment
Read more

Straight Line - Notes, Concept and All Important Formula

STRAIGHT LINE Table Of Contents 1. RELATION BETWEEN CARTESIAN CO-ORDINATE & POLAR CO-ORDINATE SYSTEM If \((x, y)\) are Cartesian co-ordinates of a point \(P\) , then : \(x=r \cos \theta\) , \(y=r \sin \theta\) and \(r=\sqrt{x^{2}+y^{2}}, \quad \theta=\tan ^{-1}\left(\dfrac{y}{x}\right)\) All Chapter Notes, Concept and Important Formula 2. DISTANCE FORMULA AND ITS APPLICATIONS : If \(\mathrm{A}\left(\mathrm{x}_{1}, \mathrm{y}_{1}\right)\) and \(\mathrm{B}\left(\mathrm{x}_{2}, \mathrm{y}_{2}\right)\) are two points, then \(\mathbf{A B=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}}}\) Note : (i) Three given points \(A, B\) and \(C\) are collinear, when sum of any two distances out of \(\mathrm{AB}, \mathrm{BC}, \mathrm{CA}\) is equal to the remaining third otherwise the points will be the vertices of triangle. (ii) Let \(A, B, C \& D\) be the four given points in a plane. Then the quadrilateral will be: (a) Square if \(A B=B C=C D=D...
Post a Comment
Read more

Definite Integration - Notes, Concept and All Important Formula

DEFINITE INTEGRATION Table Of Contents 1. (a) The Fundamental Theorem of Calculus, Part 1: If \(\mathrm{f}\) is continuous on \([\mathrm{a}, \mathrm{b}]\) , then the function \(\mathrm{g}\) defined by \(g(x)=\displaystyle \int_{a}^{x} f(t) d t,\)   \(  a \leq x \leq b\) is continuous on \([\mathrm{a}, \mathrm{b}]\) and differentiable on \((\mathrm{a}, \mathrm{b})\) , and \(g^{\prime}(\mathrm{x})=\mathrm{f}(\mathrm{x})\) . (b) The Fundamental Theorem of Calculus, Part 2: If f is continuous on \([a, b]\) , then \(\displaystyle \int_{a}^{b} f(x) d x=F(b)-F(a)\) where \(F\) is any antiderivative of \(\mathrm{f}\) , that is, a function such that \(\mathrm{F}^{\prime}=\mathrm{f}.\) Note : If \(\displaystyle \int_{a}^{b} f(x) d x=0 \Rightarrow\) then the equation \(f(x)=0\) has atleast one root lying in \((a, b)\) provided \(f\) is a continuous function in \((a, b)\) . All Chapter Notes, Concept and Important Formula 2. Representation of Definite Int...
Post a Comment
Read more