`int(sin(101x).sin^(99)x)dx` equals

Options

A. `(sin(100x)(sinx)^(100))/(100)+C`
B. `(cos(100x)(sinx)^(100))/(100)+C`
C. `(cos(100x)(cosx)^(100))/(100)+C`
D. `(cos(100x)(cosx)^(100))/(100)+C`

Explanation

To solve the integral \( \int \sin(101x) \sin^{99}(x) \, dx \), we can follow these steps:

### Step 1: Rewrite the Integral
We can express \( \sin(101x) \) as \( \sin(100x + x) \). Using the sine addition formula, we have:
\[ \sin(100x + x) = \sin(100x) \cos(x) + \cos(100x) \sin(x) \]
Thus, we can rewrite the integral as:
\[ \int \sin(101x) \sin^{99}(x) \, dx = \int (\sin(100x) \cos(x) + \cos(100x) \sin(x)) \sin^{99}(x) \, dx \]

### Step 2: Distribute the Integral
Now we can distribute the integral:
\[ = \int \sin(100x) \cos(x) \sin^{99}(x) \, dx + \int \cos(100x) \sin^{100}(x) \, dx \]

### Step 3: Simplify the Second Integral
The second integral can be simplified:
\[ \int \cos(100x) \sin^{100}(x) \, dx \]
We can use integration by parts for this integral. Let:
- \( u = \sin^{100}(x) \)
- \( dv = \cos(100x) \, dx \)

Then we differentiate and integrate:
- \( du = 100 \sin^{99}(x) \cos(x) \, dx \)
- \( v = \frac{1}{100} \sin(100x) \)

### Step 4: Apply Integration by Parts
Now we apply the integration by parts formula:
\[ \int u \, dv = uv - \int v \, du \]
Substituting in our values:
\[ = \frac{1}{100} \sin^{100}(x) \sin(100x) - \int \frac{1}{100} \sin(100x) (100 \sin^{99}(x) \cos(x)) \, dx \]
This simplifies to:
\[ = \frac{1}{100} \sin^{100}(x) \sin(100x) - \int \sin(100x) \sin^{99}(x) \cos(x) \, dx \]

### Step 5: Combine the Integrals
Now we can combine the integrals:
\[ \int \sin(100x) \sin^{99}(x) \cos(x) \, dx + \int \sin(100x) \sin^{99}(x) \cos(x) \, dx = \frac{1}{100} \sin^{100}(x) \sin(100x) \]
This gives us:
\[ 2 \int \sin(100x) \sin^{99}(x) \cos(x) \, dx = \frac{1}{100} \sin^{100}(x) \sin(100x) \]
Thus:
\[ \int \sin(100x) \sin^{99}(x) \cos(x) \, dx = \frac{1}{200} \sin^{100}(x) \sin(100x) \]

### Step 6: Final Result
Finally, we can write the result of the original integral:
\[ \int \sin(101x) \sin^{99}(x) \, dx = \frac{1}{200} \sin^{100}(x) \sin(100x) + C \]
where \( C \) is the constant of integration.