Questions: ∫ cos^5(10x) dx =

Solution Steps

To solve the integral of \(\cos^5(10x)\), we can use a trigonometric identity to express \(\cos^5(10x)\) in terms of lower powers of cosine and sine. Then, we can apply substitution to simplify the integration process.

To solve \(\int \cos^5(10x) \, dx\), we start by expressing \(\cos^5(10x)\) using trigonometric identities. We can use the identity \(\cos^2(\theta) = \frac{1 + \cos(2\theta)}{2}\) to reduce the power.

Express \(\cos^5(10x)\) as \((\cos^2(10x))^2 \cdot \cos(10x)\) and apply the identity: \[ \cos^2(10x) = \frac{1 + \cos(20x)}{2} \] Thus, \[ \cos^5(10x) = \left(\frac{1 + \cos(20x)}{2}\right)^2 \cdot \cos(10x) \]

Let \(u = \sin(10x)\), then \(du = 10\cos(10x) \, dx\) or \(dx = \frac{du}{10\cos(10x)}\). Substitute and simplify the integral.

Integrate the simplified expression: \[ \int \cos^5(10x) \, dx = \int \left(\frac{1 + \cos(20x)}{2}\right)^2 \cdot \cos(10x) \, dx \]

The integral evaluates to: \[ \frac{\sin^5(10x)}{50} - \frac{\sin^3(10x)}{15} + \frac{\sin(10x)}{10} + C \]

Final Answer