how to solve the indefinite integral of x^3/(square root of x^2 + 100)dx using x = 10tantheta

To solve the indefinite integral 

\[
\int \frac{x^3}{\sqrt{x^2 + 100}} \, dx
\]

using the substitution \( x = 10 \tan \theta \), follow these steps:

### Step 1: Substitution
Let \( x = 10 \tan \theta \). Then, the differential \( dx \) becomes:

\[
dx = 10 \sec^2 \theta \, d\theta
\]

Substitute \( x = 10 \tan \theta \) into the integral:

\[
\sqrt{x^2 + 100} = \sqrt{(10 \tan \theta)^2 + 100} = \sqrt{100 \tan^2 \theta + 100} = \sqrt{100(\tan^2 \theta + 1)}
\]

Since \( \tan^2 \theta + 1 = \sec^2 \theta \):

\[
\sqrt{x^2 + 100} = \sqrt{100 \sec^2 \theta} = 10 \sec \theta
\]

### Step 2: Rewrite the Integral
Now, the integral becomes:

\[
\int \frac{(10 \tan \theta)^3}{10 \sec \theta} \cdot 10 \sec^2 \theta \, d\theta
\]

Simplify the expression:

\[
\int \frac{1000 \tan^3 \theta}{\sec \theta} \cdot 10 \sec^2 \theta \, d\theta = 10^4 \int \frac{\tan^3 \theta \sec^2 \theta}{\sec \theta} \, d\theta = 10^4 \int \tan^3 \theta \sec \theta \, d\theta
\]

### Step 3: Simplify and Integrate
Use the identity \( \tan^2 \theta = \sec^2 \theta - 1 \):

\[
10^4 \int \tan^2 \theta \sec \theta \cdot \tan \theta \, d\theta = 10^4 \int (\sec^2 \theta - 1) \sec \theta \cdot \tan \theta \, d\theta
\]

Distribute:

\[
10^4 \int (\sec^3 \theta \tan \theta - \sec \theta \tan \theta) \, d\theta
\]

This integral can be split into two separate integrals:

\[
10^4 \left( \int \sec^3 \theta \tan \theta \, d\theta - \int \sec \theta \tan \theta \, d\theta \right)
\]

### Step 4: Solve Each Integral
1. **For** \( \int \sec^3 \theta \tan \theta \, d\theta \):
   - This integral is easier solved using \( u = \sec \theta \), giving:
     \[
     \int \sec^3 \theta \tan \theta \, d\theta = \frac{1}{4} \sec^4 \theta + C_1
     \]
   
2. **For** \( \int \sec \theta \tan \theta \, d\theta \):
   - This integral is straightforward:
     \[
     \int \sec \theta \tan \theta \, d\theta = \sec \theta + C_2
     \]

### Step 5: Combine and Back-Substitute
Combining both integrals, you have:

\[
10^4 \left(\frac{1}{4} \sec^4 \theta - \sec \theta\right) + C
\]

Back-substitute \( \theta = \tan^{-1}\left(\frac{x}{10}\right) \):

\[
\sec \theta = \sqrt{1 + \tan^2 \theta} = \sqrt{1 + \frac{x^2}{100}} = \frac{\sqrt{x^2 + 100}}{10}
\]

Thus, the solution is:

\[
\frac{10^4}{4} \left(\frac{x^2 + 100}{100}\right)^2 - 10^4 \cdot \frac{\sqrt{x^2 + 100}}{10} + C
\]

Simplify the expression to obtain the final answer:

\[
\frac{1}{4} \left(\frac{x^2 + 100}{10}\right)^2 - 1000\sqrt{x^2 + 100} + C
\]

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

### Related Questions
1. How would you solve this integral using a different substitution?
2. What if the integral had an additional term in the numerator?
3. Can you evaluate the definite integral with specific bounds?
4. How would the solution change if the integrand was \( \frac{x^2}{\sqrt{x^2 + 100}} \)?
5. How can you verify the correctness of the indefinite integral?

### Tip:
When using trigonometric substitutions, always remember to express all parts of the integral (including \(dx\)) in terms of the new variable before integrating.

Learn how to solve the indefinite integral of x^3 / sqrt(x^2 + 100) using the trigonometric substitution x = 10tan(θ). This step-by-step guide covers key calculus concepts such as integration techniques and trigonometric identities, suitable for college-level calculus students.

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