Mister Exam

Integral of sin^7x dx

from to
v

from to

The solution

You have entered [src]
  1
/
|
|     7
|  sin (x) dx
|
/
0             
$$\int\limits_{0}^{1} \sin^{7}{\left(x \right)}\, dx$$
Detail solution
1. Rewrite the integrand:

2. There are multiple ways to do this integral.

Method #1

1. Rewrite the integrand:

2. Integrate term-by-term:

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

1. Let .

Then let and substitute :

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

1. The integral of is when :

So, the result is:

Now substitute back in:

So, the result is:

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

1. Let .

Then let and substitute :

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

1. The integral of is when :

So, the result is:

Now substitute back in:

So, the result is:

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

1. Let .

Then let and substitute :

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

1. The integral of is when :

So, the result is:

Now substitute back in:

So, the result is:

1. The integral of sine is negative cosine:

The result is:

Method #2

1. Rewrite the integrand:

2. Integrate term-by-term:

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

1. Let .

Then let and substitute :

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

1. The integral of is when :

So, the result is:

Now substitute back in:

So, the result is:

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

1. Let .

Then let and substitute :

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

1. The integral of is when :

So, the result is:

Now substitute back in:

So, the result is:

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

1. Let .

Then let and substitute :

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

1. The integral of is when :

So, the result is:

Now substitute back in:

So, the result is:

1. The integral of sine is negative cosine:

The result is:

3. Add the constant of integration:

  /
|                                          5         7
|    7                3               3*cos (x)   cos (x)
| sin (x) dx = C + cos (x) - cos(x) - --------- + -------
|                                         5          7
/                                                         
$${{\cos ^7x}\over{7}}-{{3\,\cos ^5x}\over{5}}+\cos ^3x-\cos x$$
The graph
                             5         7
16      3               3*cos (1)   cos (1)
-- + cos (1) - cos(1) - --------- + -------
35                          5          7   
$${{5\,\cos ^71-21\,\cos ^51+35\,\cos ^31-35\,\cos 1}\over{35}}+{{16 }\over{35}}$$
=
=
                             5         7
16      3               3*cos (1)   cos (1)
-- + cos (1) - cos(1) - --------- + -------
35                          5          7   
$$- \cos{\left(1 \right)} - \frac{3 \cos^{5}{\left(1 \right)}}{5} + \frac{\cos^{7}{\left(1 \right)}}{7} + \cos^{3}{\left(1 \right)} + \frac{16}{35}$$
0.0488623115305527
0.0488623115305527