integrate secxdx
recall that if
u = secx, u' = secx tanx
You have
∫ secx dx
Turn that into a fraction by multiplying by
(secx + tanx)
and you have
∫ secx(secx+tanx)/(secx+tanx) dx
Now, if
u = secx+tanx,
du = secx tanx + sec^2 x
and your integral now becomes
∫ du/u = lnu = ln(secx + tanx) + C
tricky, eh?
To integrate sec(x)dx, we can use a technique called trigonometric substitution. We'll substitute sec(x) in terms of another trigonometric function using a triangle.
Step 1: Draw a right triangle with an acute angle θ and label the sides adjacent, opposite, and hypotenuse.
Step 2: Recall that sec(x) is equal to the reciprocal of cos(x), which is the ratio of the hypotenuse to the adjacent side in the right triangle. So, sec(x) = hypotenuse/adjacent.
Step 3: Let's say that the adjacent side is represented by "a". Then, the hypotenuse will be represented by "a sec(x)".
Step 4: Now, we can use the Pythagorean theorem to find the opposite side. According to Pythagoras, the square of the hypotenuse is equal to the sum of the squares of the other two sides. So, a² + opposite² = (a sec(x))².
Step 5: Solving for opposite, we get opposite = a tan(x).
Step 6: Now, we have expressed sec(x) and the other trigonometric functions in terms of the sides of the triangle. We can rewrite sec(x)dx as (a sec(x))(dx) and substitute sec(x) with (a sec(x)).
The integral of sec(x)dx now becomes the integral of (a sec(x))(dx):
∫(a sec(x))(dx)
Step 7: To make the substitution, we need to rewrite dx in terms of dθ. Since opposite = a tan(x), we can differentiate both sides to get:
d(opposite) = a sec²(x) dx
dx = d(opposite) / (a sec²(x))
Step 8: Substitute sec(x) with (a sec(x)) and dx with d(opposite) / (a sec²(x)):
∫((a sec(x))(d(opposite) / (a sec²(x)))
Step 9: Simplify the integral:
∫(d(opposite)/sec(x))
Step 10: Recalling that sec(x) = hypotenuse/adjacent, we can express sec(x) as (a sec(x))/a:
∫(d(opposite)/((a sec(x))/a))
Step 11: Simplify further:
∫(a d(opposite)/(a sec(x)))
Step 12: Cancel out the "a" terms:
∫(d(opposite)/sec(x))
Step 13: As we can see, ∫(d(opposite)/sec(x)) simplifies to ∫d(opposite). Integrating d(opposite) gives us opposite + C, where C is the constant of integration.
Therefore, the integral of sec(x)dx is equal to opposite + C, where opposite is a tan(x) and C is the constant of integration:
∫sec(x)dx = tan(x) + C