Using beta-gamma functions, prove that

2 sin 0 cos³ 0 de = 35

To prove the integral

\[
\int_0^{\frac{\pi}{2}} 2 \sin \theta \cos^3 \theta \, d\theta = \frac{3}{8},
\]

we will make use of **Beta** and **Gamma** functions, which are closely related to trigonometric integrals.

### Step 1: Beta Function
The Beta function \( B(x, y) \) is defined as:

\[
B(x, y) = 2 \int_0^{\frac{\pi}{2}} (\sin \theta)^{2x-1} (\cos \theta)^{2y-1} \, d\theta.
\]

Comparing this with our integral \( \int_0^{\frac{\pi}{2}} 2 \sin \theta \cos^3 \theta \, d\theta \), we observe the powers of sine and cosine, so we can relate this to the Beta function.

### Step 2: Rewrite the Integral in Beta Function Form
We need to express the integrand in the form of the Beta function:

\[
\int_0^{\frac{\pi}{2}} 2 \sin \theta \cos^3 \theta \, d\theta = 2 \cdot \int_0^{\frac{\pi}{2}} (\sin \theta)^1 (\cos \theta)^3 \, d\theta.
\]

This matches the form of the Beta function with \( x = 1 \) and \( y = 2 \), so we have:

\[
2 \cdot \int_0^{\frac{\pi}{2}} (\sin \theta)^{2(1)-1} (\cos \theta)^{2(2)-1} \, d\theta = 2 \cdot B(1, 2).
\]

### Step 3: Use the Relation Between Beta and Gamma Functions
The Beta function \( B(x, y) \) is related to the Gamma function \( \Gamma(x) \) by the following identity:

\[
B(x, y) = \frac{\Gamma(x) \Gamma(y)}{\Gamma(x + y)}.
\]

For \( B(1, 2) \), we substitute into this relation:

\[
B(1, 2) = \frac{\Gamma(1) \Gamma(2)}{\Gamma(1 + 2)} = \frac{\Gamma(1) \Gamma(2)}{\Gamma(3)}.
\]

Using the fact that \( \Gamma(1) = 1 \), \( \Gamma(2) = 1 \), and \( \Gamma(3) = 2 \), we get:

\[
B(1, 2) = \frac{1 \cdot 1}{2} = \frac{1}{2}.
\]

### Step 4: Final Calculation
Thus, the original integral becomes:

\[
2 \cdot B(1, 2) = 2 \cdot \frac{1}{2} = 1.
\]

Therefore, the value of the integral is:

\[
\int_0^{\frac{\pi}{2}} 2 \sin \theta \cos^3 \theta \, d\theta = 1.
\]

Let me know if you need further details or clarifications!

### 5 Related Questions:
1. What is the Gamma function and how does it generalize factorials?
2. How can Beta functions be used to evaluate other trigonometric integrals?
3. What are the properties of the Beta function, and how are they derived?
4. How do Beta and Gamma functions relate to probability distributions?
5. Can the Beta function be extended to complex numbers, and if so, how?

### Tip:
The Beta and Gamma functions are extremely useful in calculus and probability theory for solving integrals that involve powers of sine and cosine.

Using beta-gamma functions, prove that 2 sin 0 cos³ 0 de = 35.

This article provides a detailed proof of the integral 2 ∫ sin(θ)cos³(θ) dθ using Beta and Gamma functions. Learn the step-by-step process of applying these special functions in calculus to solve integrals involving trigonometric functions. Suitable for advanced calculus students.

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