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Identical expressions
sinxsqrt(cosx+ four)
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Similar expressions
sinxsqrt(cosx-4)

Integral of sinxsqrt(cosx+4) dx

The solution

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  1                         
  /                         
 |                          
 |           ____________   
 |  sin(x)*\/ cos(x) + 4  dx
 |                          
/                           
0

$$\int\limits_{0}^{1} \sqrt{\cos{\left(x \right)} + 4} \sin{\left(x \right)}\, dx$$

Integral(sin(x)*sqrt(cos(x) + 4), (x, 0, 1))

Detail solution

Let $u = \cos{\left(x \right)} + 4$ .

Then let $du = - \sin{\left(x \right)} dx$ and substitute $- du$ :

$\int \left(- \sqrt{u}\right)\, du$
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int \sqrt{u}\, du = - \int \sqrt{u}\, du$
  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: $- \frac{2 u^{\frac{3}{2}}}{3}$
Now substitute $u$ back in:

$- \frac{2 \left(\cos{\left(x \right)} + 4\right)^{\frac{3}{2}}}{3}$
Now simplify:

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

$- \frac{2 \left(\cos{\left(x \right)} + 4\right)^{\frac{3}{2}}}{3}+ \mathrm{constant}$

The answer is:

$- \frac{2 \left(\cos{\left(x \right)} + 4\right)^{\frac{3}{2}}}{3}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                                
 |                                              3/2
 |          ____________          2*(cos(x) + 4)   
 | sin(x)*\/ cos(x) + 4  dx = C - -----------------
 |                                        3        
/

$$\int \sqrt{\cos{\left(x \right)} + 4} \sin{\left(x \right)}\, dx = C - \frac{2 \left(\cos{\left(x \right)} + 4\right)^{\frac{3}{2}}}{3}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

      ____________        ___       ____________       
  8*\/ 4 + cos(1)    10*\/ 5    2*\/ 4 + cos(1) *cos(1)
- ---------------- + -------- - -----------------------
         3              3                  3

$$- \frac{8 \sqrt{\cos{\left(1 \right)} + 4}}{3} - \frac{2 \sqrt{\cos{\left(1 \right)} + 4} \cos{\left(1 \right)}}{3} + \frac{10 \sqrt{5}}{3}$$

      ____________        ___       ____________       
  8*\/ 4 + cos(1)    10*\/ 5    2*\/ 4 + cos(1) *cos(1)
- ---------------- + -------- - -----------------------
         3              3                  3

$$- \frac{8 \sqrt{\cos{\left(1 \right)} + 4}}{3} - \frac{2 \sqrt{\cos{\left(1 \right)} + 4} \cos{\left(1 \right)}}{3} + \frac{10 \sqrt{5}}{3}$$

-8*sqrt(4 + cos(1))/3 + 10*sqrt(5)/3 - 2*sqrt(4 + cos(1))*cos(1)/3

Numerical answer [src]

1.00391365502143

1.00391365502143

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

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Integral of sinxsqrt(cosx+4) dx

Limits of integration:

The graph:

Piecewise:

The solution