Calculus 3 : Integration

Study concepts, example questions & explanations for Calculus 3

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Example Questions

Example Question #31 : Integration

Evaluate the integral:

Possible Answers:

Correct answer:

Explanation:

To integrate, we can change the integrand using the double angle formula to something we can integrate:

Next, we make a substitution:

The derivative was found using the rule

Now, we rewrite the integral in terms of u, and integrate:

We used the following integration rules:

Finally, replace u with our original x term:

 

Example Question #32 : Integration

Integrate:

Possible Answers:

Correct answer:

Explanation:

To integrate, we must integrate by parts, using the formula

We now designate our u and dv, and differentiate and integrate, respectively:

The rules used are as follows:

Next, we use the above formula:

Finally, integrate:

The following rule was used to integrate:

Example Question #33 : Integration

Evaluate the following integral:

Possible Answers:

Correct answer:

Explanation:

The integrand can be simplified to make the problem much easier to solve. You must use the fact that  in order to rewrite the integrand. After rewriting, the problem is now:

Here, a simple u-substitution can be used:

Once again, the integrand can be rewritten with values of u:

The integral was taken by using the following rule:

Once u has been replaced with x, the final answer is:

 

Example Question #34 : Integration

Evaluate the integral of the following function:

Possible Answers:

Correct answer:

Explanation:

There are no apparent substitutions to simplify the integrand, but you can rewrite it as:

With this expression, it s clear that you must solve the integral by using partial fractions. To do this, two fractions must be created that add up to the integrand. Specifically, the partial fractions will be:

Next, the denominator must be cleared in order to solve for the unknowns. To do this, the entire equation must be multiplied by a common denomionator, . The resulting equation is:

Next, a system of equations must be created by equating the coefficients of the terms with the same power of x. The result is:

After solving, you find that 

With these values found, they can be plugged back into the original equation and integrated:

The final answer is:

The integrals were evaluated by using the following rule:

Example Question #34 : Integration

Evaluate the following integral:

Possible Answers:

Correct answer:

Explanation:

In this case, a u-substitution can be used to evaluate this integral. Specifically:

Next, the intergand must be rewritten in terms of u:

The integral was found by using the following rule:

Finally, u must be replaced with x, giving a final answer of:

Example Question #36 : Integration

Evaluate the following integral:

Possible Answers:

Correct answer:

Explanation:

The only apparent way to evaluate this integral is by using a trig substitution. The denominator resembles the form  which means that . Specifically:

With a value of x found, the radical portion of the denominator can be rewritten:

The final equation was solved by using the following trigonometric identity:

Now, every x must be replaced with theta in the integrand:

There is not an easy way to evaluate this integral, but it can be rewritten as:

Even though an integral was evaluated, it must be expressed in terms of x. To do this, refer back to the equation, . This can be rearranged to:

With this piece of information, a right triangle can be created with hypotenuse equal to seven and opposite side equal to x. After using the Pythagorean Theorem, , to solve for the rest of the triangle, you should find that:

 

Therefore, the final answer is:

 

 

Example Question #35 : Integration

Evaluate the integral of the following function:

Possible Answers:

Correct answer:

Explanation:

The only way to solve evaluate the integral is by splitting it into two separate, solvable integrals. Specifically, the integrand must be rewritten as:

The first integral must be rewritten as:

For the second integral, both the numerator and the denominator must be rewritten:

After combining the sums of each integral, the final answer is:

 

Example Question #36 : Integration

Evaluate the following integral:

Possible Answers:

Correct answer:

Explanation:

A u-substitution will work for evaluating the integral. Specifically:

Now, the integrand must be expressed in terms of u in order to evaluate the integral:

The integral was finally taken by using the following rule:

After replacing u with x, the final answer is:

Example Question #37 : Integration

Evaluate the following integral:

Possible Answers:

Correct answer:

Explanation:

The only way to evaluate this integral is by first expanding the integrand:

Next, the integrand must be evaluated as three simpler integrals:

The first integrand must be rewritten as:

The second integrand must be rewritten as:

Finally, the third integral simply equals x+C because of the following rule

After adding all of the smaller integrals together, the final answer is:

Example Question #40 : Integration

Evaluate the following integral:

Possible Answers:

Correct answer:

Explanation:

The only way to evaluate this integral is by doing a trig substitution. The radical portion of the denominator resembles the form  which means that . Specifically:

With a known value of x, the radical portion of the denominator can be rewritten as:

The final portion of the equation came from the following trigonometric identity:

Next, the integrand must be written entirely in terms of theta:

After simplifying, you are left with:

There is no way to evaluate this integral other than rewriting the integrand using the half-angle identity for cosine:

The new integrand can be split and evaluated as two separate integrals:

Even though an answer has been found, it must be in terms of x. Refer back to the equation, . This can be rearranged to . From this, you can find that . Additionally, by using the Pythagorean Theorem, , you can find any trigonometric function. The double-angle sine must be rewritten using the following double-angle idenetity:

Specifically,  and 

After rewriting theta in terms of x, the final answer is:

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