Mister Exam

Lang:

Other calculators

How to use it?
Integral of d{x}:
Integral of cosx
Integral of 1/(2+x^2)
Integral of x^5/(x^2+1)
Integral of e^(2*x)*cos(3*x)
Identical expressions
(3x+ two)/(x+ one)^(one / four)
(3x plus 2) divide by (x plus 1) to the power of (1 divide by 4)
(3x plus two) divide by (x plus one) to the power of (one divide by four)
(3x+2)/(x+1)(1/4)
3x+2/x+11/4
3x+2/x+1^1/4
(3x+2) divide by (x+1)^(1 divide by 4)
(3x+2)/(x+1)^(1/4)dx
Similar expressions
(3x-2)/(x+1)^(1/4)
(3x+2)/(x-1)^(1/4)

Integral of (3x+2)/(x+1)^(1/4) dx

The solution

You have entered [src]

 15             
  /             
 |              
 |   3*x + 2    
 |  --------- dx
 |  4 _______   
 |  \/ x + 1    
 |              
/               
0

$$\int\limits_{0}^{15} \frac{3 x + 2}{\sqrt[4]{x + 1}}\, dx$$

Integral((3*x + 2)/(x + 1)^(1/4), (x, 0, 15))

Detail solution

There are multiple ways to do this integral.
Method #1
1. Let $u = \sqrt[4]{x + 1}$ .
  
  Then let $du = \frac{dx}{4 \left(x + 1\right)^{\frac{3}{4}}}$ and substitute $du$ :
  
  $\int \left(12 u^{2} \left(u^{4} - 1\right) + 8 u^{2}\right)\, du$
  1. Integrate term-by-term:
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 12 u^{2} \left(u^{4} - 1\right)\, du = 12 \int u^{2} \left(u^{4} - 1\right)\, du$
      
      Rewrite the integrand:
      
      $u^{2} \left(u^{4} - 1\right) = u^{6} - u^{2}$
      
      Integrate term-by-term:
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{6}\, du = \frac{u^{7}}{7}$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- u^{2}\right)\, du = - \int u^{2}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{2}\, du = \frac{u^{3}}{3}$
      
      So, the result is: $- \frac{u^{3}}{3}$
      
      The result is: $\frac{u^{7}}{7} - \frac{u^{3}}{3}$
      
      So, the result is: $\frac{12 u^{7}}{7} - 4 u^{3}$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\int 8 u^{2}\, du = 8 \int u^{2}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int u^{2}\, du = \frac{u^{3}}{3}$
      
      So, the result is: $\frac{8 u^{3}}{3}$
    The result is: $\frac{12 u^{7}}{7} - \frac{4 u^{3}}{3}$
  Now substitute $u$ back in:
  
  $\frac{12 \left(x + 1\right)^{\frac{7}{4}}}{7} - \frac{4 \left(x + 1\right)^{\frac{3}{4}}}{3}$
Method #2
1. Rewrite the integrand:
  
  $\frac{3 x + 2}{\sqrt[4]{x + 1}} = \frac{3 x}{\sqrt[4]{x + 1}} + \frac{2}{\sqrt[4]{x + 1}}$
2. Integrate term-by-term:
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{3 x}{\sqrt[4]{x + 1}}\, dx = 3 \int \frac{x}{\sqrt[4]{x + 1}}\, dx$
    1. Let $u = \frac{1}{\sqrt[4]{x + 1}}$ .
      
      Then let $du = - \frac{dx}{4 \left(x + 1\right)^{\frac{5}{4}}}$ and substitute $du$ :
      
      $\int \left(- 4 \left(-1 + \frac{1}{u^{4}}\right)^{2} + 4 - \frac{4}{u^{4}}\right)\, du$
      
      Integrate term-by-term:
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- 4 \left(-1 + \frac{1}{u^{4}}\right)^{2}\right)\, du = - 4 \int \left(-1 + \frac{1}{u^{4}}\right)^{2}\, du$
      
      There are multiple ways to do this integral.
      
      Method #1
      
      Rewrite the integrand:
      
      $\left(-1 + \frac{1}{u^{4}}\right)^{2} = 1 - \frac{2}{u^{4}} + \frac{1}{u^{8}}$
      
      Integrate term-by-term:
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{2}{u^{4}}\right)\, du = - 2 \int \frac{1}{u^{4}}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{u^{4}}\, du = - \frac{1}{3 u^{3}}$
      
      So, the result is: $\frac{2}{3 u^{3}}$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{u^{8}}\, du = - \frac{1}{7 u^{7}}$
      
      The result is: $u + \frac{2}{3 u^{3}} - \frac{1}{7 u^{7}}$
      
      Method #2
      
      Rewrite the integrand:
      
      $\left(-1 + \frac{1}{u^{4}}\right)^{2} = \frac{u^{8} - 2 u^{4} + 1}{u^{8}}$
      
      Rewrite the integrand:
      
      $\frac{u^{8} - 2 u^{4} + 1}{u^{8}} = 1 - \frac{2}{u^{4}} + \frac{1}{u^{8}}$
      
      Integrate term-by-term:
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 1\, du = u$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{2}{u^{4}}\right)\, du = - 2 \int \frac{1}{u^{4}}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{u^{4}}\, du = - \frac{1}{3 u^{3}}$
      
      So, the result is: $\frac{2}{3 u^{3}}$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{u^{8}}\, du = - \frac{1}{7 u^{7}}$
      
      The result is: $u + \frac{2}{3 u^{3}} - \frac{1}{7 u^{7}}$
      
      So, the result is: $- 4 u - \frac{8}{3 u^{3}} + \frac{4}{7 u^{7}}$
      
      The integral of a constant is the constant times the variable of integration:
      
      $\int 4\, du = 4 u$
      
      The integral of a constant times a function is the constant times the integral of the function:
      
      $\int \left(- \frac{4}{u^{4}}\right)\, du = - 4 \int \frac{1}{u^{4}}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{u^{4}}\, du = - \frac{1}{3 u^{3}}$
      
      So, the result is: $\frac{4}{3 u^{3}}$
      
      The result is: $- \frac{4}{3 u^{3}} + \frac{4}{7 u^{7}}$
      
      Now substitute $u$ back in:
      
      $\frac{4 \left(x + 1\right)^{\frac{7}{4}}}{7} - \frac{4 \left(x + 1\right)^{\frac{3}{4}}}{3}$
    So, the result is: $\frac{12 \left(x + 1\right)^{\frac{7}{4}}}{7} - 4 \left(x + 1\right)^{\frac{3}{4}}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \frac{2}{\sqrt[4]{x + 1}}\, dx = 2 \int \frac{1}{\sqrt[4]{x + 1}}\, dx$
    1. Let $u = x + 1$ .
      
      Then let $du = dx$ and substitute $du$ :
      
      $\int \frac{1}{\sqrt[4]{u}}\, du$
      
      The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
      
      $\int \frac{1}{\sqrt[4]{u}}\, du = \frac{4 u^{\frac{3}{4}}}{3}$
      
      Now substitute $u$ back in:
      
      $\frac{4 \left(x + 1\right)^{\frac{3}{4}}}{3}$
    So, the result is: $\frac{8 \left(x + 1\right)^{\frac{3}{4}}}{3}$
  The result is: $\frac{12 \left(x + 1\right)^{\frac{7}{4}}}{7} - \frac{4 \left(x + 1\right)^{\frac{3}{4}}}{3}$
Now simplify:

$\frac{4 \left(x + 1\right)^{\frac{3}{4}} \left(9 x + 2\right)}{21}$
Add the constant of integration:

$\frac{4 \left(x + 1\right)^{\frac{3}{4}} \left(9 x + 2\right)}{21}+ \mathrm{constant}$

The answer is:

$\frac{4 \left(x + 1\right)^{\frac{3}{4}} \left(9 x + 2\right)}{21}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                               
 |                             3/4             7/4
 |  3*x + 2           4*(x + 1)      12*(x + 1)   
 | --------- dx = C - ------------ + -------------
 | 4 _______               3               7      
 | \/ x + 1                                       
 |                                                
/

$$\int \frac{3 x + 2}{\sqrt[4]{x + 1}}\, dx = C + \frac{12 \left(x + 1\right)^{\frac{7}{4}}}{7} - \frac{4 \left(x + 1\right)^{\frac{3}{4}}}{3}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

4376
----
 21

$$\frac{4376}{21}$$

4376
----
 21

$$\frac{4376}{21}$$

4376/21

Numerical answer [src]

208.380952380952

208.380952380952

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

Other calculators

How to use it?

Identical expressions

Similar expressions

Integral of (3x+2)/(x+1)^(1/4) dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2

Method #1

Method #2