1. Evaluate the indefinite integral

integral symbol[6x5+2sec(x) tan(x)]dx.

2. Integral symbol 8 at the top, 5 at the bottom 3x2+3x2 dx =

Thanks

1. To evaluate the indefinite integral of the function ∫(6x^5 + 2sec(x)tan(x))dx, you can apply the power rule for integration and the trigonometric identity for sec(x)tan(x).

Step 1: Start by integrating the term 6x^5. Applying the power rule of integration, you increase the exponent by 1 and divide by the new exponent:
∫(6x^5)dx = (6/6)x^6 + C = x^6 + C, where C is the constant of integration.

Step 2: Next, integrate the term 2sec(x)tan(x). By recognizing that sec(x)tan(x) is the derivative of sec(x), you can simplify the integral further.
∫(2sec(x)tan(x))dx = 2∫sec(x)tan(x)dx = 2sec(x) + C, where C is the constant of integration.

Step 3: Combine the results from Step 1 and Step 2 to obtain the final answer:
∫(6x^5 + 2sec(x)tan(x))dx = x^6 + 2sec(x) + C, where C is the constant of integration.

2. To evaluate the definite integral of the function ∫[8, 5] (3x^2 + 3x^2)dx, you can calculate the integral by applying the power rule for integration and evaluating the difference between the values at the upper and lower bounds.

Step 1: Integrate the term 3x^2. By applying the power rule of integration, you increase the exponent by 1 and divide by the new exponent:
∫(3x^2)dx = (3/3)x^3 = x^3

Step 2: Integrate the term 3x^2 again:
∫(3x^2)dx = (3/3)x^3 = x^3

Step 3: Combine the results from Step 1 and Step 2 to obtain the definite integral:
∫[8, 5] (3x^2 + 3x^2)dx = [x^3] evaluated from 8 to 5 + [x^3] evaluated from 8 to 5.

Step 4: Evaluate the definite integral by plugging in the upper bound (8) and subtracting the result obtained by plugging in the lower bound (5):
[x^3] evaluated from 8 to 5 + [x^3] evaluated from 8 to 5 = (8^3) - (5^3) + (8^3) - (5^3)

Simplifying the expression gives you the final result.