Use Fundamental Theorem to compute the integral exactly. \int_0^t (\sin^2x + \cos^2x)\,dx

Question:

Use Fundamental Theorem to compute the integral exactly.

{eq}\int_0^t (\sin^2x + \cos^2x)\,dx {/eq}

Fundamental Theorem of Calculus

When we wish to evaluate an integral, we may be able to do so by applying the Fundamental Theorem of Calculus. In order to do so, we need to be able to find the antiderivative of the integrand and use it in the following formula.

{eq}\int_a^b f(x) dx = F(b) - F(a) {/eq}

Answer and Explanation:

Before we attempt to apply the Fundamental Theorem of Calculus to this integral, let's examine the integrand. We need to be able to find its antiderivative. This is a difficult task for this function as it's expressed, but notice that it takes on a very familiar form. We can actually apply a Pythagorean identity to simplify this integrand.

{eq}\displaystyle \int_0^t (\sin^2x + \cos^2x)\ dx = \int_0^t 1 \ dx {/eq}


We can certainly find the antiderivative of a constant. This allows us to apply the Fundamental Theorem to evaluate this integral.

{eq}\begin{align*} \int_0^t 1 \ dx &= x |_0^t\\ &= t-0\\ &= t \end{align*} {/eq}


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The Fundamental Theorem of Calculus

from Math 104: Calculus

Chapter 12 / Lesson 10
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