Evaluate the integrals in Exercises 97–110.
97. ∫ 3x^(√3) dx

Verified step by step guidance

1. Identify the integral to be solved: \(\int 3x^{\sqrt{3}} \, dx\).
2. Recall the power rule for integration: for any real number \(n \neq -1\), \(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\).
3. Since the integrand is \(3x^{\sqrt{3}}\), factor out the constant 3: \(3 \int x^{\sqrt{3}} \, dx\).
4. Apply the power rule to \(\int x^{\sqrt{3}} \, dx\), which gives \(\frac{x^{\sqrt{3} + 1}}{\sqrt{3} + 1} + C\).
5. Multiply the result by 3 to get the integral: \(3 \times \frac{x^{\sqrt{3} + 1}}{\sqrt{3} + 1} + C\).

Key Concepts

Integration of Power Functions

Integrating power functions involves applying the rule ∫ x^n dx = (x^(n+1)) / (n+1) + C, where n ≠ -1. This formula is fundamental for integrating expressions where the variable is raised to a constant exponent.

Handling Irrational Exponents

When the exponent is an irrational number, such as √3, the integration process remains the same as with rational exponents. Treat the exponent as a constant and apply the power rule accordingly.

Constant Multipliers in Integration

Constants multiplied by functions can be factored out of the integral. For example, ∫ 3x^n dx = 3 ∫ x^n dx, simplifying the integration process by focusing on the variable part.