Integration of 1/(sin x cos x^3): Method and Solution
Integration problems involving trigonometric functions can often be tackled using substitution techniques. In this article, we will explore the process of integrating the function 1/(sin x cos x^3). This type of integral requires a combination of substitution and completing the square to obtain the final solution. We'll go through the complete process step-by-step, explaining each transformation and technique used.
Introduction to the Integral: 1/(sin x cos x^3)
Consider the function we wish to integrate:
∫(1/(sin x cos x^3)) dx
This integral can appear daunting due to the combination of sine and cosine terms, but by applying suitable substitutions and algebraic manipulations, we can achieve a solvable form. We'll utilize the tangent half-angle substitution, also known as the Weierstrass substitution.
Step-by-Step Solution
Substitution: t tan(x/2)
Start with the substitution:
t tan(x/2)
This implies:
sin x 2t / (1 t^2) cos x (1 - t^2) / (1 t^2) dx 2 / (1 t^2) dtSubstituting these into the original integral:
∫(1/(2t / (1 t^2) * (1 - t^2) / (1 t^2) * 3)) * 2 / (1 t^2) dt ∫(1 / (2t(1 - t^2) / (1 t^2) * 3)) * 2 / (1 t^2) dt ∫(1 / (2t(1 - t^2) / (1 t^2) * 3) * 2 / (1 t^2) dt ∫(1 / (2t^2 - 2t 4)) dt
Now, multiply by 4 to complete the square:
4(2t^2 - 2t 4) 8t^2 - 8t 16 8(t^2 - t/2 1) 8((t - 1/2)^2 7/4) 8(t - 1/2)^2 14
Completing the Square: u 2t - 1
Rewrite the integral in a more manageable form:
∫(4 / (8(t - 1/2)^2 14)) dt ∫(4 / ((2(t - 1/2))^2 14)) dt (2/√7) ∫(1 / (u^2 1)) du
Finally, integrate using the arctangent function:
(2/√7) arctan(u) C (2/√7) arctan((2t - 1)/√7) C
Back-Substitution: t tan(x/2)
Replace t with tan(x/2):
(2/√7) arctan((2 tan(x/2) - 1)/√7) C
Conclusion
In conclusion, the integral of 1/(sin x cos x^3) can be solved using a combination of substitution and completing the square techniques. The step-by-step process includes the Weierstrass substitution to transform the integral into a more manageable form, which can then be solved using standard integration methods. Understanding these techniques is crucial for tackling more complex trigonometric integrals in mathematics and engineering.