Mister Exam

Lang:

Other calculators

How to use it?
Integral of d{x}:
Integral of x*2^(-x)
Integral of (1/x^2)dx
Integral of x*cos(x/2)
Integral of y^(-2/3)
Identical expressions
exp(sin^2x)*sin2xdx
exponent of ( sinus of squared x) multiply by sinus of 2xdx
exp(sin2x)*sin2xdx
expsin2x*sin2xdx
exp(sin²x)*sin2xdx
exp(sin to the power of 2x)*sin2xdx
exp(sin^2x)sin2xdx
exp(sin2x)sin2xdx
expsin2xsin2xdx
expsin^2xsin2xdx

Integral of exp(sin^2x)*sin2xdx dx

The solution

You have entered [src]

  1                     
  /                     
 |                      
 |      2               
 |   sin (x)            
 |  e       *sin(2*x) dx
 |                      
/                       
0

$$\int\limits_{0}^{1} e^{\sin^{2}{\left(x \right)}} \sin{\left(2 x \right)}\, dx$$

Integral(exp(sin(x)^2)*sin(2*x), (x, 0, 1))

Detail solution

There are multiple ways to do this integral.
Method #1
1. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 2 e^{\sin^{2}{\left(x \right)}} \sin{\left(x \right)} \cos{\left(x \right)}\, dx = 2 \int e^{\sin^{2}{\left(x \right)}} \sin{\left(x \right)} \cos{\left(x \right)}\, dx$
  1. Let $u = \sin^{2}{\left(x \right)}$ .
    
    Then let $du = 2 \sin{\left(x \right)} \cos{\left(x \right)} dx$ and substitute $\frac{du}{2}$ :
    
    $\int \frac{e^{u}}{2}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\text{False}$
      
      The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
      
      So, the result is: $\frac{e^{u}}{2}$
    Now substitute $u$ back in:
    
    $\frac{e^{\sin^{2}{\left(x \right)}}}{2}$
  So, the result is: $e^{\sin^{2}{\left(x \right)}}$
Method #2
1. Rewrite the integrand:
  
  $e^{\sin^{2}{\left(x \right)}} \sin{\left(2 x \right)} = 2 e^{\sin^{2}{\left(x \right)}} \sin{\left(x \right)} \cos{\left(x \right)}$
2. The integral of a constant times a function is the constant times the integral of the function:
  
  $\int 2 e^{\sin^{2}{\left(x \right)}} \sin{\left(x \right)} \cos{\left(x \right)}\, dx = 2 \int e^{\sin^{2}{\left(x \right)}} \sin{\left(x \right)} \cos{\left(x \right)}\, dx$
  1. Let $u = \sin^{2}{\left(x \right)}$ .
    
    Then let $du = 2 \sin{\left(x \right)} \cos{\left(x \right)} dx$ and substitute $\frac{du}{2}$ :
    
    $\int \frac{e^{u}}{2}\, du$
    1. The integral of a constant times a function is the constant times the integral of the function:
      
      $\text{False}$
      
      The integral of the exponential function is itself.
      
      $\int e^{u}\, du = e^{u}$
      
      So, the result is: $\frac{e^{u}}{2}$
    Now substitute $u$ back in:
    
    $\frac{e^{\sin^{2}{\left(x \right)}}}{2}$
  So, the result is: $e^{\sin^{2}{\left(x \right)}}$
Add the constant of integration:

$e^{\sin^{2}{\left(x \right)}}+ \mathrm{constant}$

The answer is:

$e^{\sin^{2}{\left(x \right)}}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                   
 |                                    
 |     2                          2   
 |  sin (x)                    sin (x)
 | e       *sin(2*x) dx = C + e       
 |                                    
/

$$\int e^{\sin^{2}{\left(x \right)}} \sin{\left(2 x \right)}\, dx = C + e^{\sin^{2}{\left(x \right)}}$$

Simplify

The graph

Plot the graph f(x)

The answer [src]

         2   
      sin (1)
-1 + e

$$-1 + e^{\sin^{2}{\left(1 \right)}}$$

         2   
      sin (1)
-1 + e

$$-1 + e^{\sin^{2}{\left(1 \right)}}$$

-1 + exp(sin(1)^2)

Numerical answer [src]

1.03007638063326

1.03007638063326

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

Other calculators

How to use it?

Identical expressions

Integral of exp(sin^2x)*sin2xdx dx

Limits of integration:

The graph:

Piecewise:

The solution

Method #1

Method #2