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

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Question

Evaluate this indefinite integral:

$\int sin^{4}(x)cos^{3}(x)dx$

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Answer

To approach this problem, first rewrite the integral expression as shown below:

$\int sin^{4}(x)cos^{2}(x)cos(x)dx$ .

Then, recognize that , and substitute this into the integral expression:

$\int sin^{4}(x)(1-sin^{2}x)cos(x)dx$

Use substitution, letting and . The integral can then be rewritten as

$\int u^{4}(1-u^{2})du = \int (u^{4} - u^{6})du.$

Evaluating this integral gives

$\frac{u^{5}}{5} - \frac{u^{7}}{7} + C$ .

Finally, substituting back into this expression gives the final answer:

$\frac{sin^{5}(x)}{5} - \frac{sin^{7}(x)}{7} + C$

(As this is an indefinite integral, must be included).

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Question

Evaluate the following indefinite integral.

$\int (x-2x^3+1)dx$

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Answer

Use the inverse Power Rule to evaluate the integral. We know that $\int x^n dx= \frac{x^{n+1}}{n+1}+C$ for $n \ne -1$ . 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.

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Question

Evaluate the definite integral of the algebraic function.

integral (x3 + √(x))dx from 0 to 1

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Answer

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

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Question

Evaluate the integral.

Integral from 1 to 2 of (1/x3) dx

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Answer

Integral from 1 to 2 of (1/x3) dx

Integral from 1 to 2 of (x-3) dx

Integrate the integral.

from 1 to 2 of (x–2/-2)

(2–2/–2) – (1–2/–2) = (–1/8) – (–1/2)=(3/8)

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Question

Evaluate the following indefinite integral.

$\int 4 dx$

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Answer

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".

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Question

$\int \sec x\tan x\ \mathrm{d}x=$

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Answer

The answer is . The definition of the derivative of is . Remember to add the to undefined integrals.

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Question

$\int_{-1}^{0}e^{1-t}dt =$

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Answer

We can use the substitution technique to evaluate this integral.

Let .

We will differentiate with respect to .

$\frac{\mathrm{d}u }{\mathrm{d} t}=-1$ , which means that ${\mathrm{d} u}=-{\mathrm{d} t}$ .

We can solve for ${\mathrm{d}t }$ in terms of ${\mathrm{d} u}$ , which gives us ${\mathrm{d}t }=-{\mathrm{d} u}$ .

We will also need to change the bounds of the integral. When , , and when , .

We will now substitute in for the , and we will substitute $-{\mathrm{d}u }$ for ${\mathrm{d} t}$ .

$\int_{2}^{1}-e^{u}du$

$\int_{2}^{1}-e^{u}du = -e^{u}|_{2}^{1}=-e^{1}-(-e^{2})=e^{2}-e^{1}$

The answer is $e^{2}-e$ .

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Question

Evaluate the following indefinite integral.

$\int (x^3 - x^2 + 3)dx$

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Answer

Use the inverse Power Rule to evaluate the integral. We know that $\int x^n dx= \frac{x^{n+1}}{n+1}+C$ for $n \ne -1$ . 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.

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Question

Evaluate the following indefinite integral.

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

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Answer

Use the inverse Power Rule to evaluate the integral. We know that $\int x^n dx= \frac{x^{n+1}}{n+1}+C$ for $n \ne -1$ . 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.

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Question

Evaluate the following indefinite integral.

$\int 3y dy$

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Answer

Use the inverse Power Rule to evaluate the integral. Firstly, constants can be taken out of integrals, so we pull the 3 out front. Next, according to the inverse power rule, we know that $\int x^n dx= \frac{x^{n+1}}{n+1}+C$ for $n \ne -1$ . 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.

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Question

Evaluate the following indefinite integral.

$\int (1-x)dx$

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Answer

Use the inverse Power Rule to evaluate the integral. We know that $\int x^n dx= \frac{x^{n+1}}{n+1}+C$ for $n \ne -1$ . 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.

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Question

Evaluate the following indefinite integral.

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

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Answer

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 $n \ne -1$ . 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.

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Question

Evaluate the following indefinite integral.

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

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Answer

Use the inverse Power Rule to evaluate the integral. We know that $\int x^n dx= \frac{x^{n+1}}{n+1}+C$ for $n \ne -1$ . 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.

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Question

Evaluate the following indefinite integral.

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

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Answer

Use the inverse Power Rule to evaluate the integral. We know that $\int x^n dx= \frac{x^{n+1}}{n+1}+C$ for $n \ne -1$ . 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.

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Question

Evaluate the following definite integral.

$\int_{0}^{2}(x^2+3) dx$

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Answer

Unlike an indefinite integral, the definite integral must be evaluated at its limits, in this case, from 0 to 2. First, we use our inverse power rule to find the antiderivative. So, we have that $\int x^ndx=\frac{x^{n+1}}{n+1}$ . Once you find the antiderivative, we must remember that where is the indefinite integral. So, we plug in our limits and subtract the two. So, we have $(\frac{1}{3}x^3+3x)|_0^2=[\frac{1}{3}2^3+3(2)]-[\frac{1}{3}0^3+3(0)]=26/3$ .

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Question

Evaluate the following definite integral.

$\int_{1}^{3}6xdx$

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Answer

Unlike an indefinite integral, the definite integral must be evaluated at its limits, in this case, from 1 to 3. First, we use our inverse power rule to find the antiderivative. So, we have that $\int x^ndx=\frac{x^{n+1}}{n+1}$ . Once you find the antiderivative, we must remember that where is the indefinite integral. So, we plug in our limits and subtract the two. So, we have .

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Question

Evaluate the following definite integral.

$\int_{1}^{4}\frac{1}{x}dx$

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Answer

Unlike an indefinite integral, the definite integral must be evaluated at its limits, in this case, from 1 to 4. First, we use our inverse power rule to find the antiderivative. So since is to the power of , we have that $\int x^{-1}=ln|x|$ . Once you find the antiderivative, we must remember that where is the indefinite integral. So, we plug in our limits and subtract the two. So, we have because we know that $ln\ 1=0$ .

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Question

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

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Answer

The integral of $\frac{1}{x}$ is $\ln \left | x \right |+C$ . The constant 3 is simply multiplied by the integral.

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Question

Evaluate:

$\int_{1}^{5} (3x^{2} - \frac{5}{x}) dx$

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Answer

The first step is to find the antiderivative, recalling that:

$\int \frac{1}{x} dx= ln \left | x\right |$ .

For this integral:

$\int_{1}^{5} (3x^{2} - \frac{5}{x}) dx = x^{3} - 5 ln(x)$ ,

where the intergral would be evaluated from to (the absolute value bar is not necessary, since both limits of integration are greater than zero):

$5^{3} - 5 ln(5) - (1^3 - 5 ln (1) =$

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Question

Evaluate the following indefinite integral:

$\int tan^{5}(x)sec^{2}(x) dx$

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Answer

Use substitution, where $u=\tan (x)$ and $du=\sec ^{2}(x)dx$ . Thus, the integral can be rewritten as:

$\int tan^{5}(x)sec^{2}(x) dx= \int u^{5} du = \frac{u^{6}}{6} + C$ .

Substitution of $u=\tan (x)$ back into this expression gives the final answer:

$\frac{tan^{6}(x)}{6} + C$

Note that since this is an indefinite integral, the addition of a constant term (C) is required.

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