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Integral of ln(x^2+3)
Identical expressions
ln(x^ two + three)
ln(x squared plus 3)
ln(x to the power of two plus three)
ln(x2+3)
lnx2+3
ln(x²+3)
ln(x to the power of 2+3)
lnx^2+3
ln(x^2+3)dx
Similar expressions
ln(x^2-3)

Solving integrals/ ln(x^2+3)

Integral of ln(x^2+3) dx

The solution

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  1               
  /               
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 |     / 2    \   
 |  log\x  + 3/ dx
 |                
/                 
0

$$\int\limits_{0}^{1} \log{\left(x^{2} + 3 \right)}\, dx$$

Integral(log(x^2 + 3), (x, 0, 1))

Detail solution

Use integration by parts:

$\int \operatorname{u} \operatorname{dv} = \operatorname{u}\operatorname{v} - \int \operatorname{v} \operatorname{du}$

Let $u{\left(x \right)} = \log{\left(x^{2} + 3 \right)}$ and let $\operatorname{dv}{\left(x \right)} = 1$ .

Then $\operatorname{du}{\left(x \right)} = \frac{2 x}{x^{2} + 3}$ .

To find $v{\left(x \right)}$ :
1. The integral of a constant is the constant times the variable of integration:
  
  $\int 1\, dx = x$
Now evaluate the sub-integral.
The integral of a constant times a function is the constant times the integral of the function:

$\int \frac{2 x^{2}}{x^{2} + 3}\, dx = 2 \int \frac{x^{2}}{x^{2} + 3}\, dx$
1. Rewrite the integrand:
  
  $\frac{x^{2}}{x^{2} + 3} = 1 - \frac{3}{x^{2} + 3}$
2. Integrate term-by-term:
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int 1\, dx = x$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{3}{x^{2} + 3}\right)\, dx = - 3 \int \frac{1}{x^{2} + 3}\, dx$
    So, the result is: $- \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} x}{3} \right)}$
  The result is: $x - \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} x}{3} \right)}$
So, the result is: $2 x - 2 \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} x}{3} \right)}$
Now simplify:

$x \log{\left(x^{2} + 3 \right)} - 2 x + 2 \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} x}{3} \right)}$
Add the constant of integration:

$x \log{\left(x^{2} + 3 \right)} - 2 x + 2 \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} x}{3} \right)}+ \mathrm{constant}$

The answer is:

$x \log{\left(x^{2} + 3 \right)} - 2 x + 2 \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} x}{3} \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                                
 |                                                        /    ___\
 |    / 2    \                     / 2    \       ___     |x*\/ 3 |
 | log\x  + 3/ dx = C - 2*x + x*log\x  + 3/ + 2*\/ 3 *atan|-------|
 |                                                        \   3   /
/

$$\int \log{\left(x^{2} + 3 \right)}\, dx = C + x \log{\left(x^{2} + 3 \right)} - 2 x + 2 \sqrt{3} \operatorname{atan}{\left(\frac{\sqrt{3} x}{3} \right)}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

          ___         
     pi*\/ 3          
-2 + -------- + log(4)
        3

$$-2 + \log{\left(4 \right)} + \frac{\sqrt{3} \pi}{3}$$

          ___         
     pi*\/ 3          
-2 + -------- + log(4)
        3

$$-2 + \log{\left(4 \right)} + \frac{\sqrt{3} \pi}{3}$$

-2 + pi*sqrt(3)/3 + log(4)

Numerical answer [src]

1.20009372535411

1.20009372535411

The graph

Use the examples entering the upper and lower limits of integration.

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Identical expressions

Similar expressions

Integral of ln(x^2+3) dx

Limits of integration:

The graph:

Piecewise:

The solution