`int_(0)^(pi) xsin x cos^(4)x dx=`

Options

A. `(pi)/(10)`
B. `(pi)/(5)`
C. `-(pi)/(5)`
D. none of these

Answer

To solve the integral \( I = \int_{0}^{\pi} x \sin x \cos^4 x \, dx \), we can use a property of definite integrals that states:

\[ \int_{a}^{b} f(x) \, dx = \int_{a}^{b} f(a + b - x) \, dx \]

In this case, \( a = 0 \) and \( b = \pi \), so we can rewrite the integral as follows:

### Step 1: Apply the property of definite integrals

\[ I = \int_{0}^{\pi} x \sin x \cos^4 x \, dx = \int_{0}^{\pi} (\pi - x) \sin(\pi - x) \cos^4(\pi - x) \, dx \]

### Step 2: Simplify the expression

Using the identities \( \sin(\pi - x) = \sin x \) and \( \cos(\pi - x) = -\cos x \), we can simplify:

\[ I = \int_{0}^{\pi} (\pi - x) \sin x \cos^4 x \, dx \]

### Step 3: Rewrite the integral

Now we can express \( I \) as:

\[ I = \int_{0}^{\pi} \pi \sin x \cos^4 x \, dx - \int_{0}^{\pi} x \sin x \cos^4 x \, dx \]

This gives us:

\[ I = \pi \int_{0}^{\pi} \sin x \cos^4 x \, dx - I \]

### Step 4: Solve for \( I \)

Adding \( I \) to both sides:

\[ 2I = \pi \int_{0}^{\pi} \sin x \cos^4 x \, dx \]

Thus,

\[ I = \frac{\pi}{2} \int_{0}^{\pi} \sin x \cos^4 x \, dx \]

### Step 5: Evaluate the integral \( \int_{0}^{\pi} \sin x \cos^4 x \, dx \)

To evaluate this integral, we can use the substitution \( t = \cos x \), which gives \( dt = -\sin x \, dx \). The limits change as follows: when \( x = 0 \), \( t = 1 \) and when \( x = \pi \), \( t = -1 \).

Thus:

\[ \int_{0}^{\pi} \sin x \cos^4 x \, dx = \int_{1}^{-1} (-t^4) \, dt = \int_{-1}^{1} t^4 \, dt \]

### Step 6: Calculate the integral \( \int_{-1}^{1} t^4 \, dt \)

This integral can be computed as:

\[ \int_{-1}^{1} t^4 \, dt = 2 \int_{0}^{1} t^4 \, dt = 2 \left[ \frac{t^5}{5} \right]_{0}^{1} = 2 \cdot \frac{1}{5} = \frac{2}{5} \]

### Step 7: Substitute back to find \( I \)

Substituting this back into the equation for \( I \):

\[ I = \frac{\pi}{2} \cdot \frac{2}{5} = \frac{\pi}{5} \]

### Final Answer

Thus, the value of the integral is:

\[ \boxed{\frac{\pi}{5}} \]