Question

Evaluate: \( \int x^9 \cdot \sec^2(x^{10}) \, dx \).

Hint:

For integrals involving powers of \(x\) and trigonometric functions, use substitution to simplify the expression.

Answer 1

Step 1: Use substitution.
Let \( u = x^{10} \). Then, \[ \frac{du}{dx} = 10x^9 $\Rightarrow$ dx = \frac{du}{10x^9} \]

Step 2: Rewrite the integral.
Substituting into the integral: \[ \int x^9 \cdot \sec^2(x^{10}) \, dx = \int \sec^2(u) \cdot \frac{du}{10} \] \[ = \frac{1}{10} \int \sec^2(u) \, du \]

Step 3: Integrate.
The integral of \( \sec^2(u) \) is \( \tan(u) \), so: \[ \frac{1}{10} \int \sec^2(u) \, du = \frac{1}{10} \tan(u) + C \]

Step 4: Substitute back for \( u \).
Since \( u = x^{10} \), the final answer is: \[ \frac{1}{10} \tan(x^{10}) + C \]

Final Answer: \[ \boxed{\frac{1}{10} \tan(x^{10}) + C} \]