91Ó°ÊÓ

An antiderivative of a function is essentially another function that, when differentiated, gives back the original function. In other words, if you have a function \( f(x) \), and another function \( F(x) \) such that \( F'(x) = f(x) \), then \( F(x) \) is an antiderivative of \( f(x) \). This is a fundamental concept in calculus, often used to find the indefinite integral of a function.

When dealing with the indefinite integral \( \int (-5 \sin t) \, dt \), we are searching for a function whose derivative is \(-5 \sin t\). Through the process of integration, we determine that the antiderivative is \(5 \cos t\). This is found by recognizing the derivative formula for trigonometric functions, since \(-\cos t\), when differentiated, results in \(\sin t\). Therefore, \(-5 \cos t\) becomes \(5 \cos t\) after including the constant factor.

Sometimes, integrating a function requires more than just applying basic rules or recognizing common derivatives. Integration by substitution is a technique that can simplify complex integrals, making them easier to solve.
To use substitution, one typically looks for a part of the integral that can be replaced with a simpler variable \(u\). After redefining the integral in terms of \(u\), it may become possible to find an easier integral to solve. However, in our example involving \(-5 \sin t\), substitution is not necessary since \(\sin t\) is directly integrable with a simple rule.

For cases where substitution is needed, you might follow these steps:

• Choose a substitution \(u = g(t)\) such that its derivative \(du\) is present in some form in the integral.
• Change the limits of integration if necessary, if it's a definite integral.
• Rewrite the integral in terms of \(u\).
• Integrate with respect to \(u\).
• Substitute back to the original variable.

When finding the indefinite integral of a function, the result includes an arbitrary constant, usually denoted by \(C\). This constant is crucial because the antiderivative is not unique. There exists an entire family of functions that differ only by a constant after differentiation.

In our example, after integrating \(-5 \sin t\), we arrive at \(5 \cos t\). Adding \(C\) gives us the most general form: \(5 \cos t + C\). Every integration of this type should incorporate this constant of integration.

Why is this constant important?

• When differentiating, differentiating a constant yields zero, so any constant we add or subtract during integration will not affect the derivative.
• It ensures all possible antiderivatives are considered, as one function might require a specific constant \(C\) to satisfy certain initial conditions or boundary values within an applied problem context.

Including \(C\) ensures no solutions are inadvertently omitted.