Trigonometric Integrals

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GRE Quantitative Reasoning › Trigonometric Integrals

Questions 1 - 6
1

Evaluate the following integral.

Explanation

Recall: The trig identity

We can rewrite the integral using the above identity as

We can now solve the integral using substitution and

The last step is to reinsert our substitution:

2

Integrate the following.

Explanation

We can integrate the function by using substitution where so .

Just focus on integrating sine now:

The last step is to reinsert the substitution:

3

Integrate the following.

Explanation

We can integrate using substitution:

and so

Now we can just focus on integrating cosine:

Once the integration is complete, we can reinsert our substitution:

4

Fnd the derivative of tan(x) with respect to x or

Derivative cannot be found

Explanation

The is one of the trigonometric integrals that must be memorized.

Other common trig derivatives that should be memorized are:

5

Evaluate the following integral.

Explanation

Recall: The identity

The integral can be rewritten as

Because of the trig identity above, we can rewrite it in a different way:

Now we can integrate using substitution where and

Finally, we reinsert our substitution:

6

Evaluate:

Explanation

  1. The 1/2 is a constant, and so is pulled out front.

  2. The integral of cos(x) is sin(x), by definition.

  3. Writing the limits for evaluation:

  1. Using the unit circle, , and .

5)Simplifying:

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