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Integrals with unbounded integrands. Evaluate the following integral or state that it diverges....

Question:

Integrals with unbounded integrands. Evaluate the following integral or state that it diverges.

{eq}\int^{1}_{0} \ln y^{2} dy {/eq}

Definite Integral :

Here we will use the integration by parts in order to determine the given definite integral -

$$\int_{}^{} \; uv \; dx = u \; \int_{}^{} \; v \;dx - \int_{}^{} \; \frac {du}{dx} \; \biggr( \int_{}^{} \; v \; dx \biggr) \; dx $$

$$\int_{}^{} \; x^{n} \; dx = \frac {x^{n+1}}{n+1} + C $$

$$\Rightarrow \ln(x^{n}) = n \; \ln(x) $$

Answer and Explanation: 1

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$$\text {The integral which we have to determine is given as -} $$

$$I = \int_{0}^{1} \; \ln(y^{2}) \; dy = 2 \; \int_{0}^{1} \; 1 \times \ln(y) \;...

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Evaluating Definite Integrals Using the Fundamental Theorem

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Chapter 16 / Lesson 2
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The fundamental theorem of calculus makes finding your definite integral almost a piece of cake. See how the definite integral becomes a subtraction problem after applying the fundamental theorem of calculus.


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