SE \( x=\operatorname{senh} t \) \( -\frac{1}{4} \operatorname{senh} 2 t+\frac{1}{2} t+c \)

The function \( x = \operatorname{senh} t \) refers to the hyperbolic sine, which is defined as \( \operatorname{senh} t = \frac{e^t - e^{-t}}{2} \). This function is often used in mathematics, particularly in areas dealing with hyperbolic geometry or in the solutions of certain differential equations. Understanding its properties can greatly enhance your grasp of calculus and mathematical analysis. When approaching the expression \( -\frac{1}{4} \operatorname{senh} 2t + \frac{1}{2} t + c \), it’s helpful to remember that the hyperbolic identity \( \operatorname{senh} 2t = 2 \operatorname{senh} t \operatorname{cosh} t \) can simplify calculations. A common mistake here is forgetting to properly apply the chain rule when differentiating hyperbolic functions, which can lead to significant errors in solutions or calculus problems. Always double-check your derivatives!