Question

Найти первообразную функции корень из x+x^1/3

Answer 1

Первообразная    в общем виде

F(√x+x^1/3) =   2/3 x^3/2 +3/4 x^4/3

Answer 2

Finding the Antiderivative of \( \sqrt{x} \cdot x^{\frac{1}{3}} \)

To find the antiderivative of \( \sqrt{x} \cdot x^{\frac{1}{3}} \), we can use the properties of derivatives and the power rule for integration. Let's break down the process step by step.

1. Rewrite the Function: We can rewrite \( \sqrt{x} \) as \( x^{\frac{1}{2}} \). Therefore, the function becomes: \[ x^{\frac{1}{2}} \cdot x^{\frac{1}{3}} \]

2. Combine the Terms: Using the properties of exponents, we can combine the terms: \[ x^{\frac{1}{2}} \cdot x^{\frac{1}{3}} = x^{\frac{1}{2} + \frac{1}{3}} = x^{\frac{5}{6}} \]

3. Apply the Power Rule for Integration: The power rule for integration states that for any real number \( n \neq -1 \), the antiderivative of \( x^n \) with respect to \( x \) is \( \frac{x^{n+1}}{n+1} + C \), where \( C \) is the constant of integration.

Therefore, the antiderivative of \( x^{\frac{5}{6}} \) is: \[ \int x^{\frac{5}{6}} \, dx = \frac{6}{11}x^{\frac{11}{6}} + C \]

Where \( C \) is the constant of integration.

4. Final Result: The antiderivative of \( \sqrt{x} \cdot x^{\frac{1}{3}} \) is: \[ \frac{6}{11}x^{\frac{11}{6}} + C \]

Where \( C \) is the constant of integration.

Conclusion

The antiderivative of \( \sqrt{x} \cdot x^{\frac{1}{3}} \) is \( \frac{6}{11}x^{\frac{11}{6}} + C \), where \( C \) is the constant of integration.