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Integral of 10*sin^2x
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Identical expressions
ten *sin^2x
10 multiply by sinus of squared x
ten multiply by sinus of squared x
10*sin2x
10*sin²x
10*sin to the power of 2x
10sin^2x
10sin2x
10*sin^2xdx

Solving integrals/ 10*sin^2x

Integral of 10*sin^2x dx

The solution

You have entered [src]

 pi              
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 2               
  /              
 |               
 |        2      
 |  10*sin (x) dx
 |               
/                
pi               
--               
4

$$\int\limits_{\frac{\pi}{4}}^{\frac{\pi}{2}} 10 \sin^{2}{\left(x \right)}\, dx$$

Simplify

Detail solution

The integral of a constant times a function is the constant times the integral of the function:

$\int 10 \sin^{2}{\left(x \right)}\, dx = 10 \int \sin^{2}{\left(x \right)}\, dx$
1. Rewrite the integrand:
  
  $\sin^{2}{\left(x \right)} = \frac{1}{2} - \frac{\cos{\left(2 x \right)}}{2}$
2. Integrate term-by-term:
  1. The integral of a constant is the constant times the variable of integration:
    
    $\int \frac{1}{2}\, dx = \frac{x}{2}$
  1. The integral of a constant times a function is the constant times the integral of the function:
    
    $\int \left(- \frac{\cos{\left(2 x \right)}}{2}\right)\, dx = - \frac{\int \cos{\left(2 x \right)}\, dx}{2}$
    1. Let $u = 2 x$ .
      
      Then let $du = 2 dx$ and substitute $\frac{du}{2}$ :
      
      $\int \frac{\cos{\left(u \right)}}{4}\, du$
      1. The integral of a constant times a function is the constant times the integral of the function:
        
        $\int \frac{\cos{\left(u \right)}}{2}\, du = \frac{\int \cos{\left(u \right)}\, du}{2}$
        
        The integral of cosine is sine:
        
        $\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$
        
        So, the result is: $\frac{\sin{\left(u \right)}}{2}$
      Now substitute $u$ back in:
      
      $\frac{\sin{\left(2 x \right)}}{2}$
    So, the result is: $- \frac{\sin{\left(2 x \right)}}{4}$
  The result is: $\frac{x}{2} - \frac{\sin{\left(2 x \right)}}{4}$
So, the result is: $5 x - \frac{5 \sin{\left(2 x \right)}}{2}$
Add the constant of integration:

$5 x - \frac{5 \sin{\left(2 x \right)}}{2}+ \mathrm{constant}$

The answer is:

$5 x - \frac{5 \sin{\left(2 x \right)}}{2}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                    
 |                                     
 |       2                   5*sin(2*x)
 | 10*sin (x) dx = C + 5*x - ----------
 |                               2     
/

$$5\,\left(x-{{\sin \left(2\,x\right)}\over{2}}\right)$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

5   5*pi
- + ----
2    4

$$10\,\left(-{{\sin \pi-\pi}\over{4}}-{{\pi-2\,\sin \left({{\pi }\over{2}}\right)}\over{8}}\right)$$

5   5*pi
- + ----
2    4

$$\frac{5}{2} + \frac{5 \pi}{4}$$

Simplify

Numerical answer [src]

6.42699081698724

6.42699081698724

The graph

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

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Identical expressions

Integral of 10*sin^2x dx

Limits of integration:

The graph:

Piecewise:

The solution