Mister exam

Integral sin⁵x dx

from to

from to

The solution

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$$\int\limits_{0}^{1} \sin^{5}{\left(x \right)}\, dx$$
Integral(sin(x)^5, (x, 0, 1))
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. 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:

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. 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:

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:

The graph
                 5           3
8             cos (1)   2*cos (1)
-- - cos(1) - ------- + ---------
15               5          3    
$$- \cos{\left(1 \right)} - \frac{\cos^{5}{\left(1 \right)}}{5} + \frac{2 \cos^{3}{\left(1 \right)}}{3} + \frac{8}{15}$$
=
=
                 5           3
8             cos (1)   2*cos (1)
-- - cos(1) - ------- + ---------
15               5          3    
$$- \cos{\left(1 \right)} - \frac{\cos^{5}{\left(1 \right)}}{5} + \frac{2 \cos^{3}{\left(1 \right)}}{3} + \frac{8}{15}$$
8/15 - cos(1) - cos(1)^5/5 + 2*cos(1)^3/3
0.0889743964515759
0.0889743964515759
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$$\int \sin^{5}{\left(x \right)}\, dx = C - \frac{\cos^{5}{\left(x \right)}}{5} + \frac{2 \cos^{3}{\left(x \right)}}{3} - \cos{\left(x \right)}$$