Mister Exam

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Identical expressions
(sqrt(3sinx+ one))*cosx
( square root of (3 sinus of x plus 1)) multiply by co sinus of e of x
( square root of (3 sinus of x plus one)) multiply by co sinus of e of x
(√(3sinx+1))*cosx
(sqrt(3sinx+1))cosx
sqrt3sinx+1cosx
(sqrt(3sinx+1))*cosxdx
Similar expressions
9sqrt(3sinx+1)cosx
(sqrt(3sinx-1))*cosx

Integral of (sqrt(3sinx+1))*cosx dx

The solution

You have entered [src]

 pi                           
 --                           
 2                            
  /                           
 |                            
 |    ______________          
 |  \/ 3*sin(x) + 1 *cos(x) dx
 |                            
/                             
0

$$\int\limits_{0}^{\frac{\pi}{2}} \sqrt{3 \sin{\left(x \right)} + 1} \cos{\left(x \right)}\, dx$$

Integral(sqrt(3*sin(x) + 1)*cos(x), (x, 0, pi/2))

Detail solution

Let $u = 3 \sin{\left(x \right)} + 1$ .

Then let $du = 3 \cos{\left(x \right)} dx$ and substitute $\frac{du}{3}$ :

$\int \frac{\sqrt{u}}{3}\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \sqrt{u}\, du = \frac{\int \sqrt{u}\, du}{3}$
  1. The integral of $u^{n}$ is $\frac{u^{n + 1}}{n + 1}$ when $n \neq -1$ :
    
    $\int \sqrt{u}\, du = \frac{2 u^{\frac{3}{2}}}{3}$
  So, the result is: $\frac{2 u^{\frac{3}{2}}}{9}$
Now substitute $u$ back in:

$\frac{2 \left(3 \sin{\left(x \right)} + 1\right)^{\frac{3}{2}}}{9}$
Now simplify:

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

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

The answer is:

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

The answer (Indefinite) [src]

  /                                                    
 |                                                  3/2
 |   ______________                 2*(3*sin(x) + 1)   
 | \/ 3*sin(x) + 1 *cos(x) dx = C + -------------------
 |                                           9         
/

$$\int \sqrt{3 \sin{\left(x \right)} + 1} \cos{\left(x \right)}\, dx = C + \frac{2 \left(3 \sin{\left(x \right)} + 1\right)^{\frac{3}{2}}}{9}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

14/9

$$\frac{14}{9}$$

14/9

$$\frac{14}{9}$$

14/9

Numerical answer [src]

1.55555555555556

1.55555555555556

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

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Integral of (sqrt(3sinx+1))*cosx dx

Limits of integration:

The graph:

Piecewise:

The solution