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