Calculations of Basic Definite and Indefinite Integrals

Activity 8.15: Evaluate Each Definite Integral Below

Activity 8.15 requires the evaluation of several definite integrals. The problems involve polynomial, rational, and trigonometric functions with specific upper and lower limits of integration.

1. The first problem asks for the evaluation of the integral of a quadratic polynomial over the interval $[0, 1]$: $\int_0^1 (x^2 - 2x + 3)dx$

2. The second problem involves a product of a variable and a quadratic term, also integrated over the interval $[0, 1]$: $\int_0^1 x(x^2 + 1)dx$

3. The third problem requires integrating a sum containing a squared term and a term with a reciprocal square root over the interval $[1, 2]$: $\int_1^2 (x^2 + \frac{2}{\sqrt{x}})dx$

4. The fourth problem involves a variable multiplied by a binomial square, integrated over the interval $[0, 2]$: $\int_0^2 x(2 - 3x)^2 dx$

5. The fifth problem is a rational function integrated over the interval $[1, 2]$: $\int_1^2 \frac{x^2 + 1}{x^4} dx$

6. The sixth problem involves a higher-order polynomial function integrated over the interval $[-1, 2]$: $\int_{-1}^2 (x^4 - 3x^2 + 5)dx$

7. The seventh problem requires the evaluation of a polynomial involving a sixth-power term over the interval $[0, 5]$: $\int_0^5 (x^6 - 5x^4)dx$

8. The eighth problem includes the sum of a squared term and its reciprocal, integrated over the interval $[1, 2]$: $\int_1^2 (x^2 + \frac{1}{x^2})dx$

9. The ninth problem involves the integration of a basic trigonometric sine function over the interval $[0, \pi]$: $\int_0^\pi \sin(x)dx$

Unit Summary

In this unit, the primary focus has been the calculation of integrals of basic functions. The material covered includes the concepts and techniques required to solve both definite and indefinite integrals. The specific types of functions discussed in this unit are polynomials, exponential functions, and trigonometric functions. Through these exercises, the integration rules for power functions, transcendental functions, and the fundamental theorem of calculus are applied to derive exact numerical values for definite integrals and general anti-derivative expressions for indefinite integrals.

Activity 8.16 (Unit Test): Evaluate Each Integral Below

Activity 8.16 serves as a comprehensive unit test, requiring the evaluation of ten different integrals. These include both indefinite integrals (general anti-derivatives) and definite integrals (specific values).

1. The first unit test problem is an indefinite integral of a polynomial and a radical term: $\int (3x^2 - \sqrt{5x} + 2)dx$

2. The second problem is an indefinite integral involving negative powers of the variable $x$: $\int (\frac{3}{x^4} - \frac{2}{x^3})dx$

3. The third problem is a definite integral of a quadratic polynomial over the interval $[-1, 2]$: $\int_{-1}^2 (x^2 + 3x - 2)dx$

4. The fourth problem involves the indefinite integral of the product of two expressions containing power and radical terms: $\int (x^3 - 2\sqrt{x})(x - 5)dx$

5. The fifth problem is the simplest form of an indefinite integral, the integral of a constant: $\int 5dx$

6. The sixth problem involves an indefinite integral of terms with fractional exponents and a constant: $\int (x^{2/3} - 3x^{2/5} + 6)dx$

7. The seventh problem consists of the indefinite integral of a polynomial with multiple terms: $\int (6x^5 - 15x^2 - 3x)dx$

8. The eighth problem is a definite integral requiring the subtraction of a squared variable from a squared binomial over the interval $[0, 1]$: $\int_0^1 [(x + 2)^2 - x^2]dx$

9. The ninth problem asks for the general indefinite integral of a cubic polynomial: $\int (x^3 - 5x^2 + 6x)dx$

10. The tenth and final problem represents an indefinite integral combining a trigonometric function and an exponential function: $\int (2\cos(x) + 3e^x)dx$