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Integral of d{x}:
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Integral of a
Integral of x^2/(x-1)
Integral of ln(x-1)
Derivative of:
ln(x^2+4)
Graphing y =:
ln(x^2+4)
Identical expressions
ln(x^ two + four)
ln(x squared plus 4)
ln(x to the power of two plus four)
ln(x2+4)
lnx2+4
ln(x²+4)
ln(x to the power of 2+4)
lnx^2+4
ln(x^2+4)dx
Similar expressions
ln(x^2-4)

Solving integrals/ ln(x^2+4)

Integral of ln(x^2+4) dx

The solution

You have entered [src]

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0

$$\int\limits_{0}^{1} \log{\left(x^{2} + 4 \right)}\, dx$$

Simplify

Detail solution

Use integration by parts:

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

Let $u{\left(x \right)} = \log{\left(x^{2} + 4 \right)}$ and let $\operatorname{dv}{\left(x \right)} = 1$ .

Then $\operatorname{du}{\left(x \right)} = \frac{2 x}{x^{2} + 4}$ .

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:

$\int \frac{2 x^{2}}{x^{2} + 4}\, dx = 2 \int \frac{x^{2}}{x^{2} + 4}\, dx$
1. Rewrite the integrand:
  
  $\frac{x^{2}}{x^{2} + 4} = 1 - \frac{4}{x^{2} + 4}$
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:
    
    $\int \left(- \frac{4}{x^{2} + 4}\right)\, dx = - 4 \int \frac{1}{x^{2} + 4}\, dx$
    1. The integral of $\frac{1}{x^{2} + 1}$ is $\frac{\operatorname{atan}{\left(\frac{x}{2} \right)}}{2}$ .
    So, the result is: $- 2 \operatorname{atan}{\left(\frac{x}{2} \right)}$
  The result is: $x - 2 \operatorname{atan}{\left(\frac{x}{2} \right)}$
So, the result is: $2 x - 4 \operatorname{atan}{\left(\frac{x}{2} \right)}$
Now simplify:

$x \log{\left(x^{2} + 4 \right)} - 2 x + 4 \operatorname{atan}{\left(\frac{x}{2} \right)}$
Add the constant of integration:

$x \log{\left(x^{2} + 4 \right)} - 2 x + 4 \operatorname{atan}{\left(\frac{x}{2} \right)}+ \mathrm{constant}$

The answer is:

$x \log{\left(x^{2} + 4 \right)} - 2 x + 4 \operatorname{atan}{\left(\frac{x}{2} \right)}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                    
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 |    / 2    \                      /x\        / 2    \
 | log\x  + 4/ dx = C - 2*x + 4*atan|-| + x*log\x  + 4/
 |                                  \2/                
/

$$x\,\log \left(x^2+4\right)-2\,\left(x-2\,\arctan \left({{x}\over{2 }}\right)\right)$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

-2 + 4*atan(1/2) + log(5)

$${{2\,\log 5+8\,\arctan \left({{1}\over{2}}\right)-4}\over{2}}$$

-2 + 4*atan(1/2) + log(5)

$$-2 + \log{\left(5 \right)} + 4 \operatorname{atan}{\left(\frac{1}{2} \right)}$$

Simplify

Numerical answer [src]

1.46402834843732

1.46402834843732

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+4) dx

Limits of integration:

The graph:

Piecewise:

The solution