AP Calculus AB - Comparing relative magnitudes of functions and their rates of change

Evaluate the following indefinite integral.

$\int (4x^7+x^4+x)dx$

$\frac{1}{2}x^8+\frac{1}{5}x^5+\frac{1}{2}x^2+C$

${2}x^8+{5}x^5+{2}x^2+C$

$\frac{1}{7}x^7+\frac{1}{4}x^4+x+C$

$\frac{1}{2}x^8+\frac{1}{5}x^4+\frac{1}{2}x^2$

Explanation

Use the inverse Power Rule to evaluate the integral. We know that $\int x^n dx= \frac{x^{n+1}}{n+1}+C$ for . We see that this rule tells us to increase the power of by 1 and multiply by $\frac{1}{n+1}$ . Next always add your constant of integration that would be lost in the differentiation. Take the derivative of your answer to check your work.

Evaluate the following indefinite integral.

$\int \frac{1}{2}xdx$

$\frac{1}{4}x^2+C$

$\frac{1}{2}x^2+C$

$\frac{1}{2}x+C$

Explanation

Use the inverse Power Rule to evaluate the integral. Firstly, constants can be taken out of the integral, so we pull the 1/2 out front and then complete the integration according to the rule. We know that $\int x^n dx= \frac{x^{n+1}}{n+1}+C$ for . We see that this rule tells us to increase the power of by 1 and multiply by $\frac{1}{n+1}$ . Next always add your constant of integration that would be lost in the differentiation. Take the derivative of your answer to check your work.

Evaluate the following indefinite integral.

$\int (1-x)dx$

$x-\frac{1}{2}x^2+C$

$x+\frac{1}{2}x^2+C$

Explanation

Use the inverse Power Rule to evaluate the integral. We know that $\int x^n dx= \frac{x^{n+1}}{n+1}+C$ for . We see that this rule tells us to increase the power of by 1 and multiply by $\frac{1}{n+1}$ Next always add your constant of integration that would be lost in the differentiation. Take the derivative of your answer to check your work.

Evaluate the following indefinite integral.

$\int \frac{1}{x}dx$

Explanation

Use the inverse Power Rule to evaluate the integral. We know that $\int x^n dx= \frac{x^{n+1}}{n+1}+C$ for . But, in this case, IS equal to so a special condition of the rule applies. We must instead use $\int x^{-1}dx=ln|x|+C$ . Evaluate accordingly. Next always add your constant of integration that would be lost in the differentiation. Take the derivative of your answer to check your work.

Evaluate the following indefinite integral.

$\int 4 dx$

Explanation

In order to evaluate the indefinite integral, ask yourself, "what expression do I differentiate to get 4". Next, use the power rule and increase the power of by 1. To start, we have $4x^{0}$ , so in the answer we have $4x^{1}$ . Next add a constant that would be lost in the differentiation. To check your work, differentiate your answer and see that it matches "4".

Evaluate the following indefinite integral.

$\int \frac{3}{x}dx$

$\frac{1}{3}ln|x|+C$

$\frac{3}{2}x^3+C$

Explanation

Use the inverse Power Rule to evaluate the integral. We know that $\int x^n dx= \frac{x^{n+1}}{n+1}+C$ for . But, in this case, IS equal to so a special condition of the rule applies. We must instead use $\int x^{-1}dx=ln|x|+C$ . Pull the constant "3" out front and evaluate accordingly. Next always add your constant of integration that would be lost in the differentiation. Take the derivative of your answer to check your work.

Evaluate the definite integral of the algebraic function.

integral (x3 + √(x))dx from 0 to 1

11/12

10/12

5/12

Explanation

Step 1: Rewrite the problem.

integral (x3+x1/2) dx from 0 to 1

Step 2: Integrate

x4/4 + 2x(2/3)/3 from 0 to 1

Step 3: Plug in bounds and solve.

\[14/4 + 2(1)(2/3)/3\] – \[04/4 + 2(0)(2/3)/3\] = (1/4) + (2/3) = (3/12) + (8/12) = 11/12

Evaluate the following indefinite integral.

$\int 4xdx$

Explanation

First, we know that we can pull the constant "4" out of the integral, and we then evaluate the integral according to this equation:

$\int x^n dx=\frac{x^{n+1}}{n+1}+C, n e1$ . From this, we acquire the answer above. As a note, we cannot forget the constant of integration which would be lost during the differentiation.