Calculus 3 : Integration

Study concepts, example questions & explanations for Calculus 3

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Example Questions

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Example Question #51 : Integration

Possible Answers:

Correct answer:

Explanation:

To integrate, we simply expand the function and integrate:

 

 

Example Question #159 : Calculus Review

Possible Answers:

Correct answer:

Explanation:

To solve this problem, we would have to apply u-substitution: 

Substituting for u and dx, get us: 

The  terms cross out and simplifying we get: 

We can then 'isolate' the constant: 

To integrate , we have to convert it to 

Since the derivative of cos(x) is -sin(x), we would have to apply u-substitution again, but using a different variable to prevent confusion: 

Replacing for v and du gets us: 

Simplifying get us: 

Integrating gets us: 

Now substitute v = cos(u) from earlier: 

Now substitute u = 2/x from earlier as well and now we get: 

Example Question #52 : Integration

Possible Answers:

Correct answer:

Explanation:

We notice that this problem can be solved with a simple u -substitution: 

Substituting for u and dx get us: 

The  terms cross out, and simplifying we get: 

Isolating the -1 outside the integral we get: 

This is now simply: 

Substituting , from earlier gets us our final answer: 

Example Question #52 : Integration

Possible Answers:

Correct answer:

Explanation:

This problem can be solved using u-substitution:

Substituting for u and dx in the original integral gets us: 

However the factors x+1 and 2x+2 don't cross out, but we can factor 2x+2 to cross out x+1: 

Simplifying gets us: 

Isolating the 1/2 from the integral gets us: 

Substituting  from earlier, we now get: 

Example Question #53 : Integration

Possible Answers:

Correct answer:

Explanation:

 This question can be solved with a simple u- substitution; 

Plugging in u and dx in our original integral gets us: 

Isolating  on the outside, as it is a constant, gets us: 

The integration of cos(u) is sin(u)+C:

Substitute back  gets us: 

Example Question #54 : Integration

Possible Answers:

Correct answer:

Explanation:

This problem can be solved using a u-substitution: 

 

Plugging in u and dx gets us: 

The cos(x) factors cross out and simplifying get us: 

The integration of  is 

Plugging in u = sin(x) from earlier get us our final answer: 

Example Question #164 : Calculus Review

Possible Answers:

Correct answer:

Explanation:

This problem can be solved using u-substitution, but instead of crossing out terms, we are manipulating the terms: 

Plugging in u and du into the original integral gets us: 

Since we cant cross out any terms, we will substitute  in terms of u by solving for x: 

 

Now that we found our  term in terms of u, we can substitute it in our equation above: 

We can now expand and simplify from there: 

Then we can integrate from there: 

Substituting u = x-1 from earlier gets us:

 

Example Question #165 : Calculus Review

Possible Answers:

Correct answer:

Explanation:

We notice that the powers of the functions within the numerator and denominator are the same, which means we need to rewrite the function: 

We first isolate the any constants to make the problem easier: 

We then rewrite the function  into  though long division or any other method: 

Now we integrate each term: 

To find, , we apply u-substitution: 

Plugging in u and du, we get: 

The integral of  is , and we get: 

Substituting u = x-4 from earlier, we get our final answer: 

Example Question #166 : Calculus Review

Possible Answers:

Correct answer:

Explanation:

At first glance, we notice that the tangent power is odd and positive. Since it is odd, we then save a secant-tangent factor and convert the remaining factors to secants: 

Ignoring the secant- tangent factor, convert the remaining factors to secant functions: 

Using the trig identity, , convert 

Now, we are free to use u- substitution at this point: 

Substituting u and dx for the above equation, we now get: 

We can then cross out the sec(x)tan(x) factor to now get: 

Expand and simplify factors: 

Using the basic formula , we can now integrate the function: 

Now substitute the remaining u's with u = sec(x): 

Simplifying, we get: 

Example Question #167 : Calculus Review

Possible Answers:

Correct answer:

Explanation:

This question looks extremely difficult without thinking at first, but it is simple problem by breaking it into separate integrals: 

We then apply u substitution to each of the two separate integrals. starting with

 

Plugging in for u and dx get us: 

Isolate constants: 

The integration of  is 

Plugging in , we now get: 

For the second integral, , we also apply u substitution, but using a different variable to avoid confusion: 

Plugging y and dy into the integral gets us: 

which equals: 

Plugging in y = x+1, we now get: 

Combining two results together gets us our final result: 

 

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