Integrating Trigonometric functions (part 4)

We shall now proceed to integrating secx and similarly, lets refresh the formulas we should know.

\frac {d}{dx}tanx = sec^{2}x

\frac {d}{dx}secx = secxtanx

\int tanx dx = ln|secx|+c (MF15)

\int secx dx = ln|secx+tanx|+c (MF15)

\int sec^{2}x dx = tanx+c

\int sec^{3}x dx = \int secx(sec^{2}x)dx = \int secx(tan^{2}x+1)dx = \int secxtan^{2}x+secx dx
So how do we \int secxtan^{2}x dx? I’ll first rewrite it as \int (secxtanx)(tanx)dx for some insights.

We can’t adopt the \int f'(x)f(x) dx method here. So, Integration by parts?

\int (secxtanx)(tanx)~dx

= secx(tanx) - \int secx(sec^{2})~ dx

= secxtanx-\int sec^{3}x~dx

Wait! \int sec^{3}x dx again? hmmm.

So we have that

\int sec^{3}x ~dx

= \int secxtan^{2}x+secx ~dx

= secxtanx-\int sec^{3}xdx + \int secx ~dx.

Then with a bit of juggling and manipulations, we have

2\int sec^{3}x dx = secxtanx + ln|secx+tanx|+c.

I do hope this gives you some insights. You should try \int sec^{4}x dx on your own using the information here.

Comments
    pingbacks / trackbacks
    • […] (1) 3. Integrating Trigonometric Functions (2) 4. Integrating Trigonometric Functions (3) 5. Integrating Trigonometric Functions (4) 6. Integrating Trigonometric Functions (5) 7. Easiest way to remember cosec, sec, and cot […]

    Leave a Comment

    three × 4 =

    Contact Us

    CONTACT US We would love to hear from you. Contact us, or simply hit our personal page for more contact information

    Not readable? Change text. captcha txt

    Start typing and press Enter to search