Questions: Evaluate each of the following integrals. Your answers should be exact. (a) ∫ from -8 to 5 of 22 e^(0.14 x) dx= (b) ∫ from 0 to 2 of (25 x^4+9 x^2) e^(5 x^5+3 x^3) dx= Question Help: Video.

Solution Steps

To solve these integrals, we will use symbolic computation in Python with the help of the sympy library. This library allows us to perform symbolic mathematics, including integration.

1. Define the function \( 22 e^{0.14 x} \).
2. Use the integrate function to find the indefinite integral.
3. Evaluate the definite integral from -8 to 5.

1. Define the function \( (25 x^{4} + 9 x^{2}) e^{5 x^{5} + 3 x^{3}} \).
2. Use the integrate function to find the indefinite integral.
3. Evaluate the definite integral from 0 to 2.

To evaluate the integral \( \int_{-8}^{5} 22 e^{0.14 x} \, dx \), we first find the indefinite integral of the function \( 22 e^{0.14 x} \). The antiderivative of \( e^{0.14 x} \) is \( \frac{e^{0.14 x}}{0.14} \). Therefore, the antiderivative of \( 22 e^{0.14 x} \) is: \[ \int 22 e^{0.14 x} \, dx = \frac{22}{0.14} e^{0.14 x} = 157.1428571 e^{0.14 x} \] Next, we evaluate this antiderivative at the bounds \( x = 5 \) and \( x = -8 \): \[ \left[ 157.1428571 e^{0.14 x} \right]_{-8}^{5} = 157.1428571 \left( e^{0.14 \cdot 5} - e^{0.14 \cdot (-8)} \right) \] Calculating the values: \[ 157.1428571 \left( e^{0.7} - e^{-1.12} \right) \approx 265.1743 \]

To evaluate the integral \( \int_{0}^{2} (25 x^{4} + 9 x^{2}) e^{5 x^{5} + 3 x^{3}} \, dx \), we recognize that the integrand is of the form \( f'(x) e^{f(x)} \), where \( f(x) = 5 x^{5} + 3 x^{3} \). The antiderivative of \( f'(x) e^{f(x)} \) is \( e^{f(x)} \). Therefore: \[ \int (25 x^{4} + 9 x^{2}) e^{5 x^{5} + 3 x^{3}} \, dx = e^{5 x^{5} + 3 x^{3}} \] Next, we evaluate this antiderivative at the bounds \( x = 2 \) and \( x = 0 \): \[ \left[ e^{5 x^{5} + 3 x^{3}} \right]_{0}^{2} = e^{5 \cdot 2^{5} + 3 \cdot 2^{3}} - e^{5 \cdot 0^{5} + 3 \cdot 0^{3}} = e^{160 + 24} - e^{0} = e^{184} - 1 \] Calculating the value: \[ e^{184} - 1 \approx 8.1318 \times 10^{79} \]

Final Answer