Mister exam

# Integral cos³xsin²x dx

from to

from to

### The solution

You have entered [src]
  1
/
|
|     3       2
|  cos (x)*sin (x) dx
|
/
0                     
$$\int\limits_{0}^{1} \sin^{2}{\left(x \right)} \cos^{3}{\left(x \right)}\, dx$$
Integral(cos(x)^3*sin(x)^2, (x, 0, 1))
Detail solution
1. Rewrite the integrand:

2. There are multiple ways to do this integral.

## Method #1

1. Let .

Then let and substitute :

1. Integrate term-by-term:

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:

1. The integral of is when :

The result is:

Now substitute back in:

## 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 is when :

Now substitute back in:

So, the result is:

1. Let .

Then let and substitute :

1. The integral of is when :

Now substitute back in:

The result is:

## Method #3

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 is when :

Now substitute back in:

So, the result is:

1. Let .

Then let and substitute :

1. The integral of is when :

Now substitute back in:

The result is:

3. Add the constant of integration:

The graph
     5         3
sin (1)   sin (1)
- ------- + -------
5         3   
$$- \frac{\sin^{5}{\left(1 \right)}}{5} + \frac{\sin^{3}{\left(1 \right)}}{3}$$
=
=
     5         3
sin (1)   sin (1)
- ------- + -------
5         3   
$$- \frac{\sin^{5}{\left(1 \right)}}{5} + \frac{\sin^{3}{\left(1 \right)}}{3}$$
-sin(1)^5/5 + sin(1)^3/3
0.114230426366362
0.114230426366362
  /
/                                            
$$\int \sin^{2}{\left(x \right)} \cos^{3}{\left(x \right)}\, dx = C - \frac{\sin^{5}{\left(x \right)}}{5} + \frac{\sin^{3}{\left(x \right)}}{3}$$