Question

Evaluate the integration​

Answer 1

Topic:

Integration

Solution:

We have to evaluate the following integral.

$\displaystyle\rm I = \int\dfrac{x}{4 + x^2}\, dx$

Can also be rewritten as,

$\displaystyle\rm I = \dfrac12\int\dfrac{2x}{4 + x^2}\, dx$

Now, we will use the following rule,

$\blue{\boxed{\tt \int\dfrac{f'(x) dx}{f(x)} = \log|f(x)| + C}}$

Therefore, using this, we get:

$\displaystyle\rm I = \dfrac12\int\dfrac{d(4 + x^2)}{4 + x^2}$

$\displaystyle\rm I = \dfrac12\log|4 + x^2| + C$

Option (d) None of these is the correct answer.

More to know:

$\sf\circ\quad\int x^n\ dx = \begin{cases}\sf\dfrac{x^{n+1}}{n+1} + C, if\ n\not = -1\\\\\sf\log|x| + C, if \ n = -1\end{cases}$

$\sf\circ\quad\int e^x \ dx = e^x + C$

$\sf\circ\quad\int \log(x) \ dx = x\log(x) -x +C$

$\sf\circ\quad\int \cos(x) \ dx = \sin(x) + C$

$\sf\circ\quad\int \sin(x) \ dx = -\cos(x) + C$

$\sf\circ\quad\int \sin(ax + b) \ dx = -\dfrac{\cos(x)}{a} + C$

$\sf\circ\quad\int \cos(ax + b) \ dx = \dfrac{\sin(x)}{a} + C$