Integral Variable Exponents 516D52

1. **State the problem:** We need to find the indefinite integral $$\int (x^{23} - 3x^{x} + 3x^{2x} - 5x^{2x}) \, dx$$. 2. **Analyze each term:** - The term $$x^{23}$$ is a power function with a constant exponent. - The terms $$x^{x}$$ and $$x^{2x}$$ are more complex because the exponent depends on $$x$$. 3. **Integrate the simple power term:** For $$x^{23}$$, use the power rule: $$\int x^{n} \, dx = \frac{x^{n+1}}{n+1} + C$$ for $$n \neq -1$$. So, $$\int x^{23} \, dx = \frac{x^{24}}{24} + C$$. 4. **Consider the terms with variable exponents:** - $$x^{x}$$ and $$x^{2x}$$ do not have elementary antiderivatives expressible in terms of elementary functions. 5. **Combine like terms:** Note that $$3x^{2x} - 5x^{2x} = -2x^{2x}$$. 6. **Write the integral as:** $$\int x^{23} \, dx - 3 \int x^{x} \, dx - 2 \int x^{2x} \, dx$$. 7. **Final expression:** $$\frac{x^{24}}{24} - 3 \int x^{x} \, dx - 2 \int x^{2x} \, dx + C$$. Since $$\int x^{x} \, dx$$ and $$\int x^{2x} \, dx$$ cannot be expressed in elementary functions, the integral is expressed in terms of these integrals. **Answer:** $$\int (x^{23} - 3x^{x} + 3x^{2x} - 5x^{2x}) \, dx = \frac{x^{24}}{24} - 3 \int x^{x} \, dx - 2 \int x^{2x} \, dx + C$$.