Calculate The Given Integral 2sec6 11x Dx
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Calculating the integral of 2sec(6x+11) dx requires understanding trigonometric substitution and integration techniques. This guide provides a step-by-step approach to solving this integral, along with an online calculator for quick results.
How to Calculate the Integral
The integral of 2sec(6x+11) dx involves trigonometric substitution. Here's the step-by-step process:
- Identify the inner function: u = 6x + 11
- Find the derivative: du/dx = 6 → du = 6dx → dx = du/6
- Substitute into the integral: ∫2sec(u) * (du/6) = (2/6)∫sec(u) du
- Integrate secant: ∫sec(u) du = ln|sec(u) + tan(u)| + C
- Substitute back: (1/3)ln|sec(6x+11) + tan(6x+11)| + C
This process transforms the original integral into a solvable form using standard trigonometric identities.
The Formula
Integration Formula
∫sec(ax+b) dx = (1/a)ln|sec(ax+b) + tan(ax+b)| + C
This formula is derived from trigonometric substitution and is essential for solving integrals involving secant functions.
Worked Example
Let's calculate ∫2sec(6x+11) dx step by step:
- Let u = 6x + 11 → du = 6dx → dx = du/6
- Substitute: ∫2sec(u) * (du/6) = (1/3)∫sec(u) du
- Integrate: (1/3)ln|sec(u) + tan(u)| + C
- Substitute back: (1/3)ln|sec(6x+11) + tan(6x+11)| + C
The final result is (1/3)ln|sec(6x+11) + tan(6x+11)| + C.
Interpreting the Result
The result represents the antiderivative of the given function. The natural logarithm of the sum of secant and tangent functions captures the growth rate of the original function.
Key Points
- The result is expressed in terms of natural logarithms
- The constant of integration (C) accounts for any initial conditions
- The function grows as the argument of the secant increases
FAQ
What is the integral of secant?
The integral of secant is ln|sec(x) + tan(x)| + C. This comes from trigonometric substitution and integration by parts.
How do I handle integrals with coefficients?
Factor out the coefficient and integrate the remaining function, then multiply by the coefficient. For example, ∫2sec(x) dx = 2∫sec(x) dx.
What if the argument is more complex?
Use substitution to simplify the argument. For ∫sec(6x+11) dx, let u = 6x + 11 and proceed with the substitution method.