Use substitution to integrate. \int \frac{t^2}{9 + t^2} dt

Question:

Use substitution to integrate.

{eq}\int \frac{t^2}{9 + t^2} dt {/eq}

Definite Integral in Calculus:

The definite integral of {eq}f(t) {/eq} from {eq}a {/eq} to {eq}b {/eq} is {eq}\displaystyle \int_{a}^{b} f(t) \ dt {/eq}.

To solve this problem, we'll use integration by substitution and use the integral sum rule, which states that:

{eq}\displaystyle \int \left( f(t) + g(t) \right) \, \mathrm{d}t = \int f(t) \ dt + \int g(t) \, \mathrm{d}t {/eq}.

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We are given:

{eq}\displaystyle \int \frac{t^2}{9 + t^2} dt {/eq}

{eq}= \displaystyle \int \frac{9+t^2-9}{9 + t^2} dt {/eq}

{eq}= \displaystyle... Evaluating Definite Integrals Using the Fundamental Theorem

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Chapter 16 / Lesson 2
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In calculus, the fundamental theorem is an essential tool that helps explain the relationship between integration and differentiation. Learn about evaluating definite integrals using the fundamental theorem, and work examples to gain understanding.