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

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 ...

Mathematical Reasoning - Notes, Concept and All Important Formula

MATHEMATICAL REASONING 1. STATEMENT : A declarative sentence which is either true or false but not both, is called a statement. A sentence which is an exclamatory or a wish or an imperative or an interrogative can not be a statement. If a statement is true then its truth value is \(T\) and if it is false then its truth value is \(F\) . All Chapter Notes, Concept and Important Formula 2. SIMPLE STATEMENT: Any statement whose truth value does not depend on other statement are called simple statement. 3. COMPOUND STATEMENT : A statement which is a combination of two or more simple statements are called compound statement. Here the simple statements which form a compound statement are known as its sub statements. 4. LOGICAL CONNECTIVES : The words or phrases which combines simple statements to form a compound statement are called logical connectives. \(\scriptsize{ \begin{array}{|l|l|l|l|l|} \hline \text { S.N. } & \text { Connectives } & \text { Symbol } & \text { Use } ...

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