Mister Exam

Lang:

Other calculators

How to use it?
Integral of d{x}:
Integral of x^5dx
Integral of x/(x^2+2x+2)
Integral of x*exp(-4*x)
Integral of 1/(x^2-2*x)
Identical expressions
log(three +e^(five *x))
logarithm of (3 plus e to the power of (5 multiply by x))
logarithm of (three plus e to the power of (five multiply by x))
log(3+e(5*x))
log3+e5*x
log(3+e^(5x))
log(3+e(5x))
log3+e5x
log3+e^5x
log(3+e^(5*x))dx
Similar expressions
log(3-e^(5*x))

Integral of log(3+e^(5*x)) dx

The solution

You have entered [src]

  0                 
  /                 
 |                  
 |     /     5*x\   
 |  log\3 + E   / dx
 |                  
/                   
0

$$\int\limits_{0}^{0} \log{\left(e^{5 x} + 3 \right)}\, dx$$

Integral(log(3 + E^(5*x)), (x, 0, 0))

Detail solution

Use integration by parts:

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

Let $u{\left(x \right)} = \log{\left(e^{5 x} + 3 \right)}$ and let $\operatorname{dv}{\left(x \right)} = 1$ .

Then $\operatorname{du}{\left(x \right)} = \frac{5 e^{5 x}}{e^{5 x} + 3}$ .

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:

$\int \frac{5 x e^{5 x}}{e^{5 x} + 3}\, dx = 5 \int \frac{x e^{5 x}}{e^{5 x} + 3}\, dx$
1. Don't know the steps in finding this integral.
  
  But the integral is
  
  $\int \frac{x e^{5 x}}{e^{5 x} + 3}\, dx$
So, the result is: $5 \int \frac{x e^{5 x}}{e^{5 x} + 3}\, dx$
Now simplify:

$x \log{\left(e^{5 x} + 3 \right)} - 5 \int \frac{x e^{5 x}}{e^{5 x} + 3}\, dx$
Add the constant of integration:

$x \log{\left(e^{5 x} + 3 \right)} - 5 \int \frac{x e^{5 x}}{e^{5 x} + 3}\, dx+ \mathrm{constant}$

The answer is:

$x \log{\left(e^{5 x} + 3 \right)} - 5 \int \frac{x e^{5 x}}{e^{5 x} + 3}\, dx+ \mathrm{constant}$

The answer (Indefinite) [src]

                              /                             
  /                          |                              
 |                           |     5*x                      
 |    /     5*x\             |  x*e               /     5*x\
 | log\3 + E   / dx = C - 5* | -------- dx + x*log\3 + E   /
 |                           |      5*x                     
/                            | 3 + e                        
                             |                              
                            /

$$\int \log{\left(e^{5 x} + 3 \right)}\, dx = C + x \log{\left(e^{5 x} + 3 \right)} - 5 \int \frac{x e^{5 x}}{e^{5 x} + 3}\, dx$$

Simplify

The answer [src]

$$0$$

Numerical answer [src]

0.0

0.0

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

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of log(3+e^(5*x)) dx

Limits of integration:

The graph:

Piecewise:

The solution