University Integral Calculus Problem Solving Quiz

The integral of 3x^2 + 2x - 5 with respect to x is x^3 + x^2 - 5x + C. The integral of a polynomial is found by applying the power rule, which involves increasing the exponent of each term by 1 and dividing by the new exponent. The constant of integration, C, is added because the derivative of a constant is always zero, so there are infinitely many possible antiderivatives.

The integral of sin^2(x)cos(x) with respect to x is (1/3)sin^3(x) + C. This is a simple example of integration by substitution. We can substitute u = sin(x) and du = cos(x)dx, which transforms the integral to ∫u^2 du. This integrates to (1/3)u^3 + C, which we then substitute to find the final answer.

If f(x) = ∫ (e^(2x) + 3x) dx and f(0) = 4, then f(1) = 4e^2 - 10. Start by finding the indefinite integral of the given integrand. Then, use the initial condition, f(0) = 4, to solve for the constant of integration, C. Finally, evaluate the function at x = 1 to find f(1).

When converting from binary to octal, 3 binary digits are equivalent to one octal digit. This conversion is based on grouping binary digits into sets of three, starting from the least significant bit.

In the hexadecimal number system, the base of the system is 16. This means that it uses sixteen distinct symbols, usually the digits from 0 to 9 and the letters A to F, to represent values.

The value of 101101_2 when converted to decimal is 57. To convert from binary to decimal, assign a power of 2 to each digit, starting from the rightmost digit with 2^0. Then, multiply each digit by its corresponding power of 2 and add the results.

When converting from binary to hexadecimal, 4 binary digits are equivalent to one hexadecimal digit. This is because each hexadecimal digit represents 16 different values, which can be expressed using four binary digits.

The value of 110110_2 when converted to octal is 26_8. To convert from binary to octal, group the binary digits in threes, starting from the rightmost digit. Then, convert each group to its octal equivalent.

The value of 10001100_2 when converted to decimal is 156_10. To convert from binary to decimal, assign a power of 2 to each digit, starting from the rightmost digit with 2^0. Then, multiply each digit by its corresponding power of 2 and add the results.