Let's solve the integral step by step: \[ \int_{0}^{10} \left[\(x) + \sin(x^2) + \sin(x3)\right]\,dx \] We can split the integral: \[ \int_{0}^{10} \sin(x)\,dx + \int_{0}^{10} \sin(x^2)\,dx + \int_{0}^{10} \sin(x^3)\,dx \] Let's evaluate each part. --- ## 1. \(\int_{0}^{10} \sin(x)\,dx\) The antiderivative of \(\sin(x)\) is \(-\cos(x)\): \[ \int_{0}^{10} \sin(x) dx = \left[-\cos(x)\right]_{0}^{10} = -\cos(10) + \cos(0) \] Since \(\cos(0) = 1\): \[ = 1 - \cos(10) \] --- ## 2. \(\int_{0}^{10} \sin(x^2)\,dx\) There is no elementary antiderivative for \(\sin(x^2)\). This integral is expressed in terms of the Fresnel S function: \[ \int \sin(x^2) dx = \frac{\sqrt{\pi}}{2\sqrt{2}} \cdot \text{FresnelS}\left(\sqrt{\frac{2}{\pi}}\,x\right) + C \] But for definite integrals, we usually use a numerical value. Numerically: \[ I_2 = \int_{0}^{10} \sin(x^2) dx \approx 0.349993 \] --- ## 3. \(\int_{0}^{10} \sin(x^3)\,dx\) Similarly, there is no elementary antiderivative. This integral is also evaluated numerically: \[ I_3 = \int_{0}^{10} \sin(x^3) dx \approx 0.094489 \] --- ## 4. Combine All Results The final answer is: \[ \int_{0}^{10} \left[\sin(x) + \sin(x^2) + \sin(x^3)\right]\,dx = 1 - \cos(10) + I_2 + I_3 \] Plug in the numbers: - \(\cos(10) \approx -0.8390715\) - \(I_2 \approx 0.349993\) - \(I_3 \approx 0.094489\) \[ = 1 - (-0.8390715) + 0.349993 + 0.094489 \] \[ = 1 + 0.8390715 + 0.349993 + 0.094489 \] \[ = 2.283554 \] --- ## **Final Answer** \[ \boxed{2.28} \] (Rounded to two decimal places)