Sec(x) Integration Made Easy Using Trigonometric Substitution

secx dx

To integrate sec(x) dx, you can use the method of trigonometric substitution

To integrate sec(x) dx, you can use the method of trigonometric substitution.

1. Start by rewriting sec(x) as 1/cos(x).

2. Let’s introduce a new variable, t, and substitute t for cos(x) in the integral.

t = cos(x)
dt = -sin(x) dx

3. Rearrange the equation to solve for dx:

dx = -dt / sin(x)

4. Substitute dx and cos(x) with the corresponding expressions in terms of t:

-dt / sin(x) = -dt / sqrt(1 – t^2)

5. Now, let’s rewrite the original integral in terms of t:

∫ sec(x) dx = ∫ (1/cos(x)) dx = ∫ (1/t) (-dt / sqrt(1 – t^2))

6. Simplify the integral:

= -∫ dt / (t * sqrt(1 – t^2))

7. To integrate this, use a trigonometric substitution. Let’s substitute t with sin(u):

t = sin(u)
dt = cos(u) du

8. Rewrite the integral in terms of u:

= -∫(cos(u) du) / (sin(u) * sqrt(1 – sin^2(u)))
= -∫(cos(u) du) / (sin(u) * sqrt(cos^2(u)))
= -∫ cos(u) / (sin(u) * |cos(u)|) du

9. Simplify the absolute value of cos(u):

|cos(u)| = cos(u) when 0 ≤ u ≤ π/2
|cos(u)| = -cos(u) when π/2 < u ≤ π 10. Divide the integral into two parts depending on the range of u: = -∫ cos(u) / (sin(u) * cos(u)) du when 0 ≤ u ≤ π/2 -∫ cos(u) / (sin(u) * (-cos(u))) du when π/2 < u ≤ π 11. Simplify each integral: = -∫ du / sin(u) when 0 ≤ u ≤ π/2 ∫ du / sin(u) when π/2 < u ≤ π 12. Solve the integrals using the natural log identity: ∫ du / sin(u) = ln|csc(u) + cot(u)| + C 13. Substitute back u with the original variable x: = -ln|csc(x) + cot(x)| + C1 when 0 ≤ x ≤ π/2 ln|csc(x) + cot(x)| + C2 when π/2 < x ≤ π 14. Simplify the final result with the appropriate constants: -ln|csc(x) + cot(x)| + C when 0 ≤ x ≤ π/2 ln|csc(x) + cot(x)| + C when π/2 < x ≤ π So, the integral of sec(x) dx is: -ln|csc(x) + cot(x)| + C when 0 ≤ x ≤ π/2 ln|csc(x) + cot(x)| + C when π/2 < x ≤ π where C is the constant of integration.

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