Standard Integrals

[Anthonyk2013]

Confused as to what rules I use in these questions.

 

[ksdhart]

There are certainly other ways to solve these integrals, but I would solve these two by converting the trig functions to their sine and cosine components. For 9a:

\(\displaystyle 5\int cot\left(2t\right)\cdot csc\left(2t\right)dt=5\int \frac{cos\left(2t\right)}{sin\left(2t\right)}\cdot\frac{1}{sin\left(2t\right)}dt=5\int \frac{cos\left(2t\right)}{sin^2\left(2t\right)}dt\)

Use u-substitution, where u = sin(2t). Then for 9b:

\(\displaystyle \frac{4}{3}\int sec\left(4t\right)\cdot tan\left(4t\right)dt=\frac{4}{3}\int \frac{1}{cos\left(4t\right)}\cdot \frac{sin\left(4t\right)}{cos\left(4t\right)}dt= \frac{4}{3}\int \frac{sin\left(4t\right)}{cos^2\left(4t\right)}dt\)

Again, use u-substitution here. What would you choose as your u?

 

[Anthonyk2013]

There are certainly other ways to solve these integrals, but I would solve these two by converting the trig functions to their sine and cosine components. For 9a:

\(\displaystyle 5\int cot\left(2t\right)\cdot csc\left(2t\right)dt=5\int \frac{cos\left(2t\right)}{sin\left(2t\right)}\cdot\frac{1}{sin\left(2t\right)}dt=5\int \frac{cos\left(2t\right)}{sin^2\left(2t\right)}dt\)

Use u-substitution, where u = sin(2t). Then for 9b:

\(\displaystyle \frac{4}{3}\int sec\left(4t\right)\cdot tan\left(4t\right)dt=\frac{4}{3}\int \frac{1}{cos\left(4t\right)}\cdot \frac{sin\left(4t\right)}{cos\left(4t\right)}dt= \frac{4}{3}\int \frac{sin\left(4t\right)}{cos^2\left(4t\right)}dt\)

Again, use u-substitution here. What would you choose as your u?

Are these trig identities? its been a long time since iv do them

 

[ksdhart]

Yes, I used trig identities to rewrite the integrals. It looks to me as if you're getting into the portion of the book where you'll be working with integrals of trig functions. Memorizing some of the identities will help you greatly. Your book might have a list of helpful identities, or there are numerous "cheat sheets" you can find online to aid you in learning them.

 

[stapel]

Confused as to what rules I use in these questions.

In (9)(a), you've got a cotangent and a cosecant. Do you see any integral formula (from your table) that includes both of these? Maybe try using that formula. Use the same method for (9)(b).

If you get stuck, please reply showing your thoughts and efforts so far. Thank you!

 

[Anthonyk2013]

In (9)(a), you've got a cotangent and a cosecant. Do you see any integral formula (from your table) that includes both of these? Maybe try using that formula. Use the same method for (9)(b).

If you get stuck, please reply showing your thoughts and efforts so far. Thank you!

Its been a long day (that's my excuse anyway) formula right in front of me.