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

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 \(x\)...

Complex Number - Notes, Concept and All Important Formula

COMPLEX NUMBER 1. DEFINITION : Complex numbers are defined as expressions of the form \(a + ib\) where \(a, b \in R \quad \& i=\sqrt{-1}\) . It is denoted by \(z\) i.e. \(z=a+i b\) . 'a' is called real part of \(z(a=R e z)\) and ' \(b\) ' is called imaginary part of \(z(b=\operatorname{Im} z)\) Note : (i) The set \(R\) of real numbers is a proper subset of the Complex Numbers. Hence the Complex Number system is \(N \subset W \subset I \subset Q \subset R \subset C\) (ii) Zero is both purely real as well as purely imaginary but not imaginary. (iii) \(i =\sqrt{-1}\) is called the imaginary unit. Also \(i ^{2}=-1 ;\, i ^{3}=- i\) ; \(i ^{4}=1\) etc. (iv) \(\sqrt{a} \sqrt{b}=\sqrt{a b}\) only if atleast one of a or \(b\) is non-negative. All Chapter Notes, Concept and Important Formula 2. CONJUGATE COMPLEX : If \(z=a+i b\) then its conjugate complex is obtained by changing the sign of its imaginary part \(\&\) is denoted by \(\bar{z}\) . i.e. \(\bar{...

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...

MATHEMATICAL WORLD

Search This Blog