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

Trigonometry Ratios and Identities - Notes, Concept and All Important Formula

TRIGONOMETRIC RATIOS & IDENTITIES Table Of Contents 1. RELATION BETWEEN SYSTEM OF MEASUREMENT OF ANGLES : \(\dfrac{D}{90}=\dfrac{G}{100}=\dfrac{2 C}{\pi}\) 1 Radian \(=\dfrac{180}{\pi}\) degree \(\approx 57^{\circ} 17^{\prime} 15^{\prime \prime}\) (approximately) 1 degree \(=\dfrac{\pi}{180}\) radian \(\approx 0.0175\) radian All Chapter Notes, Concept and Important Formula 2. BASIC TRIGONOMETRIC IDENTITIES : (a) \(\sin ^{2} \theta+\cos ^{2} \theta=1\) or \(\sin ^{2} \theta=1-\cos ^{2} \theta\) or \(\cos ^{2} \theta=1-\sin ^{2} \theta\) (b) \(\sec ^{2} \theta-\tan ^{2} \theta=1\) or \(\sec ^{2} \theta=1+\tan ^{2} \theta\) or \(\tan ^{2} \theta=\sec ^{2} \theta-1\) (c) If \(\sec \theta+\tan \theta\) \(=\mathrm{k} \Rightarrow \sec \theta-\tan \theta\) \(=\dfrac{1}{\mathrm{k}} \Rightarrow 2 \sec \theta\) \(=\mathrm{k}+\dfrac{1}{\mathrm{k}}\) (d) \(\operatorname{cosec}^{2} \theta-\cot ^{2} \theta=1\) or \(\operatorname{cosec}^{2} \theta=1+\cot ^{2} \th...

Difference and relation between Differentiation and Integration

Relation between Differentiation and Integration Table Of Contents Look at the information given below. \[\mathbf{ y=f(x)}\] \[ \mathbf{ f'(x)\rightarrow \text{Derivatives of f(x)}}\] \[ \mathbf{\displaystyle \int_a^b f’(x) = ?}\] Can you tell me the value of above integral? Yes, it will be equal to f(b)-f(a) . We have already known this result. It tells us that integration is just the reverse of differentiation, integral of the derivative of the function f(x) is just equal to the difference in the function f(x) evaluated at the limits of integration. Indefinite integration- Notes and Formula Part 1 Now with this topic, we will understand how to apply this result to find the integral of a function. Consider this function \(\mathbf{g(x)=x^2}\) Let's find the integral of this function from a to b i.e \(\mathbf{\displaystyle \int_a^b g(x) \, dx}\) . Can you think how we can apply this result to find ...

Sequence And Series - Notes, Concept and All Important Formula

SEQUENCE & SERIES 1. ARITHMETIC PROGRESSION (AP) : AP is sequence whose terms increase or decrease by a fixed number. This fixed number is called the common difference . If ‘a’ is the first term & ‘d’ is the common difference, then AP can be written as a, a + d, a + 2d, ..., a + (n – 1) d, ... (a) \(n^{\text {th }}\) term of this AP \(\boxed{T_{n}=a+(n-1) d}\) , where \(d=T_{n}-T_{n-1}\) (b) The sum of the first \(n\) terms : \(\boxed{S_{n}=\frac{n}{2}[2 a+(n-1) d]=\frac{n}{2}[a+\ell]}\) , where \(\ell\) is the last term. (c) Also \(n ^{\text {th }}\) term \(\boxed{T _{ n }= S _{ n }- S _{ n -1}}\) Note: (i) Sum of first n terms of an A.P. is of the form \(A n^{2}+B n\) i.e. a quadratic expression in n, in such case the common difference is twice the coefficient of \(n ^{2}\) . i.e. 2A (ii) \(n ^{\text {th }}\) term of an A.P. is of the form \(An + B\) i.e. a linear expression in \(n\) , in such case the coefficient of \(n\) is the common difference of the ...

MATHEMATICAL WORLD

Search This Blog