Evaluate the definite integral \int_{-5}^1 \frac{1}{\sqrt{9-2x}} \, dx using appropriate...


Evaluate the definite integral {eq}\int_{-5}^1 \frac{1}{\sqrt{9-2x}} \, dx {/eq} using appropriate substitution.

Definite Integral:

We will use the substitution method to solve the integral where we will replace the part in the square root to t and then integrate with respect to variable t and then plug-in the bounds.

Answer and Explanation: 1

To solve the problem we will use the substitution method:

{eq}\int_{-5}^{1}\frac{dx}{\sqrt{9-2x}} {/eq}

Now let us put:

{eq}9-2x=t\\ -2dx=dt {/eq}

Now the integral becomes:

{eq}=\frac{-1}{2}\int t^{\frac{-1}{2}}dt\\ =-\sqrt{t}\\ =-\sqrt{9-2x} {/eq}

Now we will plug-in the bounds:

{eq}=\sqrt{19}-\sqrt{7} {/eq}

Learn more about this topic:

Evaluating Definite Integrals Using the Fundamental Theorem


Chapter 16 / Lesson 2

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.

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