## Step 1: Integrate Each Term We need to integrate the function \( 14.715 + 18\sin(10x) \): \[ \int (14.715 + 18\sin(10x)) \, dx = \int 14.715 \ dx + \int 18\sin(10x) \, dx \] - The integral of \( 14.715 \) with respect to \( x \) is \( 14.715x \). - The integral of \( 18\sin(10x) \) is \( -\frac{18}{10}\cos(10x) = -1.8\cos(10x) \), using the formula \( \int \sin(ax) dx = -\frac{1}{a}\cos(ax) \). So, the antiderivative is: \[ 14.715x - 1.8\cos(10x) \] **Explanation:** We integrated each term separately using basic integration rules. --- ## Step 2: Evaluate the Definite Integral from \( x = \) to \( x = 3 \) Plug in the bounds: \[ \left[ 14.715x - 1.8\cos(10x) \right]_^3 \] \[ = \left( 14.715 \times 3 - 1.8\cos(30) \right) - \left( 14.715 \times - 1.8\cos() \right) \] \[ = (44.145 - 1.8\cos(30)) - ( - 1.8 \times 1) \] \[ = 44.145 - 1.8\cos(30) + 1.8 \] \[ = 44.145 + 1.8 - 1.8\cos(30) \] \[ = 45.945 - 1.8\cos(30) \] Since \(\cos(30) = \cos(30^\circ) = \frac{\sqrt{3}}{2} \approx .866\): \[ 1.8 \times .866 \approx 1.5588 \] So, \[ 45.945 - 1.5588 = 44.3862 \] --- ## Final Answer \[ \boxed{44.39} \] (Rounded to two decimal places) **Explanation:** We substituted the upper and lower limits into the antiderivative, simplified using the value of \(\cos(30)\), and calculated the final value.

## Step 1: Integrate Each Term We need to integrate the function \( 14.715 + 18\sin(10x) \): \[ \int (14.715 + 18\sin(10x)) \, dx = \int 14.715 \, dx + \int 18\sin(10x) \, dx \] - The integral of \( 14.715 \) with respect to \( x \) is \( 14.715x \). - The integral of \( 18\sin(10x) \) is \( -\frac{18}{10}\cos(10x) = -1.8\cos(10x) \), using the formula \( \int \sin(ax) \, dx = -\frac{1}{a}\cos(ax) \). Thus, the antiderivative is: \[ 14.715x - 1.8\cos(10x) \] **Explanation:** We integrated each term separately using basic integration rules. --- ## Step 2: Evaluate the Definite Integral from \( x = 0 \) to \( x = 3 \) Now we evaluate the definite integral: \[ \left[ 14.715x - 1.8\cos(10x) \right]_{0}^{3} \] Substituting the bounds: \[ = \left( 14.715 \times 3 - 1.8\cos(30) \right) - \left( 14.715 \times 0 - 1.8\cos(0) \right) \] Calculating \( \cos(30) = \frac{\sqrt{3}}{2} \approx 0.866 \) and \( \cos(0) = 1 \): \[ = \left( 44.145 - 1.8 \times 0.866 \right) - (0 - 1.8 \times 1) \] \[ = 44.145 - 1.5588 + 1.8 \] \[ = 44.145 + 0.2412 \] \[ = 44.3862 \] --- ## Final Answer \[ \boxed{44.39} \] **Explanation:** We substituted the upper and lower limits into the antiderivative, simplified the expression, and calculated the final value, rounding to two decimal places.