Sign In
Ask Question
Jairo Solis
Mathematics
26 May, 21:38
How to integrate sec^3x
+2
Answers (
1
)
Rodney Villa
26 May, 21:41
0
1 Use Integration by Parts on / int / sec^{3}x /, dx∫sec
3
xdx.
Let u=/sec{x}u=secx, dv=/sec^{2}xdv=sec
2
x, du=/sec{x}/tan{x} /, dxdu=secxtanxdx, v=/tan{x}v=tanx
2 Substitute the above into uv-/int v /, duuv-∫vdu.
/sec{x}/tan{x}-/int / tan^{2}x/sec{x} /, dxsecxtanx-∫tan
2
xsecxdx
3 Use Pythagorean Identities: / tan^{2}x=/sec^{2}x-1tan
2
x=sec
2
x-1.
/sec{x}/tan{x}-/int (/sec^{2}x-1) / sec{x} /, dxsecxtanx-∫ (sec
2
x-1) secxdx
4 Expand (/sec^{2}x-1) / sec{x} (sec
2
x-1) secx.
/sec{x}/tan{x}-/int / sec^{3}x-/sec{x} /, dxsecxtanx-∫sec
3
x-secxdx
5 Use Sum Rule: / int f (x) + g (x) /, dx=/int f (x) /, dx+/int g (x) /, dx∫f (x) + g (x) dx=∫f (x) dx+∫g (x) dx.
/sec{x}/tan{x}-/int / sec^{3}x /, dx+/int / sec{x} /, dxsecxtanx-∫sec
3
xdx+∫secxdx
6 Set it as equal to the original integral / int / sec^{3}x /, dx∫sec
3
xdx.
/int / sec^{3}x /, dx=/sec{x}/tan{x}-/int / sec^{3}x /, dx+/int / sec{x} /, dx∫sec
3
xdx=secxtanx-∫sec
3
xdx+∫secxdx
7 Add / int / sec^{3}x /, dx∫sec
3
xdx to both sides.
/int / sec^{3}x /, dx+/int / sec^{3}x /, dx=/sec{x}/tan{x}+/int / sec{x} /, dx∫sec
3
xdx+∫sec
3
xdx=secxtanx+∫secxdx
8 Simplify / int / sec^{3}x /, dx+/int / sec^{3}x /, dx∫sec
3
xdx+∫sec
3
xdx to 2/int / sec^{3}x /, dx2∫sec
3
xdx.
2/int / sec^{3}x /, dx=/sec{x}/tan{x}+/int / sec{x} /, dx2∫sec
3
xdx=secxtanx+∫secxdx
9 Divide both sides by 22.
/int / sec^{3}x /, dx=/frac{/sec{x}/tan{x}+/int / sec{x} /, dx}{2}∫sec
3
xdx=
2
secxtanx+∫secxdx
10 Original integral solved.
/frac{/sec{x}/tan{x}+/int / sec{x} /, dx}{2}
2
secxtanx+∫secxdx
11 Use Trigonometric Integration: the integral of / sec{x}secx is / ln{ (/sec{x}+/tan{x}) }ln (secx+tanx).
/frac{/sec{x}/tan{x}+/ln{ (/sec{x}+/tan{x}) }}{2}
2
secxtanx+ln (secx+tanx)
12 Add constant.
/frac{/sec{x}/tan{x}+/ln{ (/sec{x}+/tan{x}) }}{2}+C
2
secxtanx+ln (secx+tanx)
+ C
Comment
Complaint
Link
Know the Answer?
Answer
Not Sure About the Answer?
Find an answer to your question 👍
“How to integrate sec^3x ...”
in 📗 Mathematics if the answers seem to be not correct or there’s no answer. Try a smart search to find answers to similar questions.
Search for Other Answers
You Might be Interested in
Eighteen less than three times a number is twice the number
Answers (1)
What is 23.791 rounded to the nearest tenth?
Answers (1)
2/4 divided by 3/10 equals
Answers (1)
Water weighs about 8.34 pounds per ounces per gallon is the weight of water
Answers (1)
Doria swan the 100-meter backstroke in 58.329 second. How do you express this time in words. show your work:
Answers (1)
New Questions in Mathematics
Identify the graph of the inequality from the given description. x is negative.
Answers (1)
What's two plus two minus five divided by eleven
Answers (2)
Solve for x. - 3x + 6 = 2x - 24 A) - 30 B) - 6 C) 6 D) 30
Answers (2)
How to identify the vertex of a quadratic function
Answers (1)
3/8 of the students in Ms. Mull's class ride the bus. If there are 24 students in the class, how many students ride the bus?
Answers (1)
Home
»
Mathematics
» How to integrate sec^3x
Sign In
Sign Up
Forgot Password?