PauloE said:
The integral of sec(x)tan(x) is ln|sec(x)| + C.
To solve the integral of sec(x)tan(x), use the substitution method by letting u = sec(x) and du = sec(x)tan(x)dx. The integral then becomes ∫du, which is simply u + C. Substituting back in for u, the final answer is ln|sec(x)| + C.
Yes, there is a shortcut known as the "secant-tangent shortcut." It involves rewriting the integral as ∫sec(x)tan(x)dx = ∫sec(x)sec(x)tan(x)dx, and then using the trigonometric identity sec^2(x) = 1 + tan^2(x). This simplifies the integral to ∫sec(x)(1 + tan^2(x))dx, which can be solved using the substitution method.
Yes, the integral of sec(x)tan(x) can be solved using integration by parts. It involves letting u = sec(x) and dv = tan(x)dx, and then using the formula ∫u(x)v'(x)dx = u(x)v(x) − ∫v(x)u'(x)dx. The integral can also be solved using integration by parts twice.
The integral of sec(x)tan(x) is used in various fields of science and engineering, such as physics, electrical engineering, and computer science. It is used to calculate the center of mass of a uniformly distributed wire, the moment of inertia of a rotating rod, and the power in an AC circuit. It is also used in the study of oscillatory motion and signal processing.