Evaluate the following integrals: `intcos^3xsin^(4)dx`

Explanation

To evaluate the integral \( \int \cos^3 x \sin^4 x \, dx \), we can follow these steps:

### Step 1: Rewrite the integral
We start with the integral:
\[ \int \cos^3 x \sin^4 x \, dx \]
We can express \( \cos^3 x \) in terms of \( \sin x \) using the identity \( \cos^2 x = 1 - \sin^2 x \):
\[ \cos^3 x = \cos x (1 - \sin^2 x) \]
Thus, we can rewrite the integral as:
\[ \int \cos x (1 - \sin^2 x) \sin^4 x \, dx \]

### Step 2: Substitution
Let \( u = \sin x \). Then, the derivative \( du = \cos x \, dx \). This means \( dx = \frac{du}{\cos x} \). Substituting these into the integral gives:
\[ \int (1 - u^2) u^4 \, du \]

### Step 3: Expand the integrand
Now, we expand the integrand:
\[ (1 - u^2) u^4 = u^4 - u^6 \]
Thus, the integral becomes:
\[ \int (u^4 - u^6) \, du \]

### Step 4: Integrate term by term
Now we can integrate each term separately:
\[ \int u^4 \, du - \int u^6 \, du \]
The integrals are:
\[ \frac{u^5}{5} - \frac{u^7}{7} + C \]

### Step 5: Substitute back
Now we substitute back \( u = \sin x \):
\[ \frac{\sin^5 x}{5} - \frac{\sin^7 x}{7} + C \]

### Step 6: Combine the results
To combine the results, we can take the common denominator:
\[ \frac{7 \sin^5 x - 5 \sin^7 x}{35} + C \]

### Final Result
Thus, the evaluated integral is:
\[ \int \cos^3 x \sin^4 x \, dx = \frac{7 \sin^5 x - 5 \sin^7 x}{35} + C \]

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