### All Calculus 3 Resources

## Example Questions

### Example Question #51 : Integration

**Possible Answers:**

**Correct answer:**

To integrate, we simply expand the function and integrate:

### Example Question #159 : Calculus Review

**Possible Answers:**

**Correct answer:**

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:**

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:**

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:**

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:**

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

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**Correct answer:**

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:**

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

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**Correct answer:**

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:**

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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