Questions: Evaluate the indefinite integral given below. [ int frac14sqrt49-4 x^2 d x ]

Solution Steps

To evaluate the indefinite integral, we can use a trigonometric substitution. Specifically, we can use the substitution \( x = \frac{7}{2} \sin(\theta) \), which simplifies the integrand into a form that is easier to integrate.

We are given the integral to evaluate: \[ \int \frac{14}{\sqrt{49 - 4x^2}} \, dx \]

To simplify the integrand, we use the substitution \( x = \frac{7}{2} \sin(\theta) \). This substitution transforms the integral into a form that is easier to integrate.

Using the substitution \( x = \frac{7}{2} \sin(\theta) \), we have: \[ dx = \frac{7}{2} \cos(\theta) \, d\theta \] The integrand becomes: \[ \frac{14}{\sqrt{49 - 4 \left(\frac{7}{2} \sin(\theta)\right)^2}} \cdot \frac{7}{2} \cos(\theta) \, d\theta \] Simplifying inside the square root: \[ 49 - 4 \left(\frac{7}{2} \sin(\theta)\right)^2 = 49 - 49 \sin^2(\theta) = 49 (1 - \sin^2(\theta)) = 49 \cos^2(\theta) \] Thus, the integrand simplifies to: \[ \frac{14}{7 \cos(\theta)} \cdot \frac{7}{2} \cos(\theta) \, d\theta = 7 \, d\theta \]

The integral now simplifies to: \[ \int 7 \, d\theta = 7\theta + C \]

We need to express \(\theta\) in terms of \(x\). From the substitution \( x = \frac{7}{2} \sin(\theta) \), we have: \[ \sin(\theta) = \frac{2x}{7} \] Thus: \[ \theta = \arcsin\left(\frac{2x}{7}\right) \] Substituting back, we get: \[ 7 \theta + C = 7 \arcsin\left(\frac{2x}{7}\right) + C \]

Final Answer