Integral of sinxd(sinx) dx

The solution

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 |  sin(x)*d*sin(x) dx
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$$\int\limits_{0}^{1} d \sin{\left(x \right)} \sin{\left(x \right)}\, dx$$

Integral(sin(x)*d*sin(x), (x, 0, 1))

Detail solution

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

$\int d \sin{\left(x \right)} \sin{\left(x \right)}\, dx = d \int \sin{\left(x \right)} \sin{\left(x \right)}\, dx$
1. Don't know the steps in finding this integral.
  
  But the integral is
  
  $\frac{x}{2} - \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{2}$
So, the result is: $d \left(\frac{x}{2} - \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{2}\right)$
Now simplify:

$\frac{d \left(2 x - \sin{\left(2 x \right)}\right)}{4}$
Add the constant of integration:

$\frac{d \left(2 x - \sin{\left(2 x \right)}\right)}{4}+ \mathrm{constant}$

The answer is:

$\frac{d \left(2 x - \sin{\left(2 x \right)}\right)}{4}+ \mathrm{constant}$

The answer (Indefinite) [src]

  /                                              
 |                            /x   cos(x)*sin(x)\
 | sin(x)*d*sin(x) dx = C + d*|- - -------------|
 |                            \2         2      /
/

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

Simplify

The answer [src]

  /1   cos(1)*sin(1)\
d*|- - -------------|
  \2         2      /

$$-{{\left(\sin 2-2\right)\,d}\over{4}}$$

  /1   cos(1)*sin(1)\
d*|- - -------------|
  \2         2      /

$$d \left(- \frac{\sin{\left(1 \right)} \cos{\left(1 \right)}}{2} + \frac{1}{2}\right)$$

Simplify

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

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Integral of sinxd(sinx) dx

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The solution