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

Permutation and Combination - Notes, Concept and All Important Formula

PERMUTATION & COMBINATION 1. FUNDAMENTAL PRINCIPLE OF COUNTING (counting without actually counting): If an event can occur in 'm' different ways, following which another event can occur in 'n' different ways, then the total number of different ways of (a) Simultaneous occurrence of both events in a definite order is \(m\) \(× n\) . This can be extended to any number of events (known as multiplication principle). (b) Happening of exactly one of the events is \(m + n\) (known as addition principle). All Chapter Notes, Concept and Important Formula 2. FACTORIAL: A Useful Notation \(: n != n ( n -1)( n -2) \ldots \ldots\) . \(2.1\) ; \(n !=n .(n-1) !\) where \(n \in W\) \(0 !=1 !=1\) \((2 n) !=2^{n} \cdot n ![1.3 .5 .7 \ldots \ldots . .(2 n-1)]\) Note that : (i) Factorial of negative integers is not defined. (ii) Let \(p\) be a prime number and \(n\) be a positive integer, then exponent of \(p\) in \(n !\) is denoted by \(E _{ p }( n !)\) and is given by \(...

Differential equations - Notes, Concept and All Important Formula

DIFFERENTIAL EQUATION 1. DIFFERENTIAL EQUATION: An equation that involves independent and dependent variables and the derivatives of the dependent variables is called a DIFFERENTIAL EQUATION. All Chapter Notes, Concept and Important Formula 2. SOLUTION (PRIMITIVE) OF DIFFERENTIAL EQUATION : Finding the unknown function which satisfies given differential equation is called SOLVING OR INTEGRATING the differential equation. The solution of the differential equation is also called its PRIMITIVE, because the differential equation can be regarded as a relation derived from it. 3. ORDER OF DIFFERENTIAL EQUATION: The order of a differential equation is the order of the highest order differential coefficient occurring in it. 4. DEGREE OF DIFFERENTIAL EQUATION: The degree of a differential equation which can be written as a polynomial in the derivatives, is the degree of the derivative of the highest order occurring in it, after it has been expressed in a form free from radicals & fracti...

Binomial Theorem - Notes, Concept and All Important Formula

BINOMIAL THEOREM \((x+y)^{n}={ }^{n} C_{0} x^{n}+{ }^{n} C_{1} x^{n-1} y+{ }^{n} C_{2} x^{n-2} y^{2}+\ldots\) \( . .+{ }^{n} C_{r} x^{n-r} y^{r}+\ldots\) \( . .+{ }^{n} C_{n} y^{n}\) \(=\displaystyle \sum_{ r =0}^{ n }{ }^{ n } C _{ r } x ^{ n - r } y ^{ r }\) , where \(n \in N\) . 1. IMPORTANT TERMS IN THE BINOMIAL EXPANSION ARE : (a) General term: The general term or the \((r+1)^{\text {th }}\) term in the expansion of \((x+y)^{n}\) is given by \(T _{ r +1}={ }^{ n } C _{ r } x ^{ n \cdot r } \cdot y ^{r}\) (b) Middle term : The middle term (s) is the expansion of \((x+y)^{n}\) is (are) : (i) If \(n\) is even, there is only one middle term which is given by \(T _{( n +2) / 2}={ }^{ n } C _{ n / 2} \cdot x ^{ n / 2} \cdot y ^{ n / 2}\) (ii) If \(n\) is odd, there are two middle terms which are \(T _{( n +1) / 2}\) & \(T _{[( n +1) / 2]+1}\) (c) Term independent of x : Term independent of \(x\) contains no \(x\) ; Hence find the value of r for which the exponent of \(x\)...

MATHEMATICAL WORLD

Search This Blog