We are asked to evaluate the definite integral:

\[
I = \int_0^7 \frac{dx}{(49 + x^2)^{\frac{3}{2}}}
\]

This type of integral suggests the use of a trigonometric substitution. The form \((a^2 + x^2)\) usually calls for the substitution \(x = a \tan \theta\), where \(a = 7\) in this case. Let's proceed step-by-step.

### Step 1: Substitution
Let:
\[
x = 7 \tan \theta \quad \text{so that} \quad dx = 7 \sec^2 \theta \, d\theta
\]

Also, note that:
\[
49 + x^2 = 49 + 49 \tan^2 \theta = 49 \sec^2 \theta
\]

Thus, the integral becomes:
\[
I = \int \frac{7 \sec^2 \theta \, d\theta}{(49 \sec^2 \theta)^{\frac{3}{2}}}
\]

### Step 2: Simplify the expression
We know that:
\[
(49 \sec^2 \theta)^{\frac{3}{2}} = 49^{\frac{3}{2}} \sec^3 \theta = 343 \sec^3 \theta
\]

Thus, the integral simplifies to:
\[
I = \int \frac{7 \sec^2 \theta \, d\theta}{343 \sec^3 \theta} = \frac{1}{49} \int \frac{d\theta}{\sec \theta} = \frac{1}{49} \int \cos \theta \, d\theta
\]

### Step 3: Integrate
The integral of \(\cos \theta\) is straightforward:
\[
\int \cos \theta \, d\theta = \sin \theta
\]

Thus, we have:
\[
I = \frac{1}{49} \sin \theta + C
\]

### Step 4: Back-substitute \(\theta\)
From the substitution \(x = 7 \tan \theta\), we know that:
\[
\tan \theta = \frac{x}{7} \quad \text{so} \quad \sin \theta = \frac{x}{\sqrt{49 + x^2}}
\]

Thus, the integral becomes:
\[
I = \frac{1}{49} \cdot \frac{x}{\sqrt{49 + x^2}} + C
\]

### Step 5: Evaluate the definite integral
Now, we evaluate the definite integral from 0 to 7:

\[
I = \left[ \frac{1}{49} \cdot \frac{x}{\sqrt{49 + x^2}} \right]_0^7
\]

At \(x = 7\):
\[
\frac{7}{\sqrt{49 + 7^2}} = \frac{7}{\sqrt{49 + 49}} = \frac{7}{\sqrt{98}} = \frac{7}{7\sqrt{2}} = \frac{1}{\sqrt{2}}
\]

At \(x = 0\):
\[
\frac{0}{\sqrt{49 + 0^2}} = 0
\]

Thus, the definite integral is:
\[
I = \frac{1}{49} \left( \frac{1}{\sqrt{2}} - 0 \right) = \frac{1}{49\sqrt{2}}
\]

So, the final answer is:
\[
I = \frac{1}{49\sqrt{2}}
\]

Would you like further clarification or more details on any part of the solution?

Here are 5 related questions you can explore:
1. How do you derive trigonometric substitution formulas?
2. How would the integral change if the exponent were different?
3. How does the substitution \(x = a \tan \theta\) help simplify integrals?
4. What happens when we integrate \(\frac{dx}{(a^2 + x^2)^{n}}\) for other values of \(n\)?
5. What if we were to change the limits of integration to different values?

**Tip:** For integrals involving expressions like \(a^2 + x^2\), trigonometric substitution is often the key to simplifying them.

Explore the step-by-step solution to the definite integral ∫[0 to 7] dx/(49+x^2)^(3/2) using trigonometric substitution. This guide covers integral calculus techniques, including substitution of x = 7 tan(θ) and simplification through trigonometric identities. Suitable for college-level calculus students.

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