How do you integrate $\int {sec}^{6} (3 x)$ $int sec^6(3x)$ ?

Question

How do you integrate $\int {sec}^{6} (3 x)$ $int sec^6(3x)$ ?

1 Answer

Impact of this question

4654 views around the world

You can reuse this answer
Creative Commons License

Answer 1 · 2017-01-10T22:11:58

$tan (3 x) + \frac{2}{3} {tan}^{3} (3 x) + \frac{1}{5} {tan}^{5} (3 x) + C$ $tan(3x)+2/3tan^3(3x)+1/5tan^5(3x)+C$

Explanation:

First let $u = 3 x$ $u=3x$ so that $d u = 3 d x$ $du=3dx$ . Then we have:

$I = \int {sec}^{6} (3 x) d x = \frac{1}{3} \int {sec}^{6} (3 x) 3 d x = \frac{1}{3} \int {sec}^{6} (u) d u$ $I=intsec^6(3x)dx=1/3intsec^6(3x)3dx=1/3intsec^6(u)du$

We will now make use of the identity ${tan}^{2} (θ) + 1 = {sec}^{2} (θ)$ $tan^2(theta)+1=sec^2(theta)$ .

$I = \frac{1}{3} \int {sec}^{4} (u) {sec}^{2} (u) d u = \frac{1}{3} \int {({sec}^{2} (u))}^{2} {sec}^{2} (u) d u$ $I=1/3intsec^4(u)sec^2(u)du=1/3int(sec^2(u))^2sec^2(u)du$

$I = \frac{1}{3} \int {(1 + {tan}^{2} (u))}^{2} {sec}^{2} (u) d u$ $I=1/3int(1+tan^2(u))^2sec^2(u)du$

Now we should perform the substitution $v = tan (u)$ $v=tan(u)$ . Note that $d v = {sec}^{2} (u) d u$ $dv=sec^2(u)du$ , which is why we left a ${sec}^{2} (u)$ $sec^2(u)$ floating around in the integrand.

$I = \int {(1 + v^{2})}^{2} d v$ $I=int(1+v^2)^2dv$

Expand ${(1 + v^{2})}^{2} = (1 + v^{2}) (1 + v^{2})$ $(1+v^2)^2=(1+v^2)(1+v^2)$ and then integrate term by term:

$I = \int (1 + 2 v^{2} + v^{4}) d v = v + \frac{2}{3} v^{3} + \frac{1}{5} v^{5}$ $I=int(1+2v^2+v^4)dv=v+2/3v^3+1/5v^5$

From $v = tan (u)$ $v=tan(u)$ :

$I = tan (u) + \frac{2}{3} {tan}^{3} (u) + \frac{1}{5} {tan}^{5} (u)$ $I=tan(u)+2/3tan^3(u)+1/5tan^5(u)$

From $u = 3 x$ $u=3x$ :

$I = tan (3 x) + \frac{2}{3} {tan}^{3} (3 x) + \frac{1}{5} {tan}^{5} (3 x) + C$ $I=tan(3x)+2/3tan^3(3x)+1/5tan^5(3x)+C$

Science

Math

Humanities

... and beyond

How do you integrate $\int {sec}^{6} (3 x)$ $int sec^6(3x)$ ?

1 Answer

Explanation:

Related questions

Impact of this question

Science

Math

Humanities

... and beyond

How do you integrate int sec^6(3x)∫sec6(3x)int sec^6(3x)?

1 Answer

Explanation:

Related questions

Impact of this question

How do you integrate $\int {sec}^{6} (3 x)$ $int sec^6(3x)$ ?