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Integral Calculus

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Question : 41 of 73

Marks: +1, -0

Evaluate

\int\limits_{0}^{2} \;\frac{x^{2} dx}{x^{2}+(2-x)^{2}}

1
0
2
None

Solution:

Concept:

Use the property

\int_0^a f(x)dx = \int_0^a f(a-x)dx

to simplify.

Explanation:

Let

I = \int_0^2 \frac{x^2}{x^2+(2-x)^2} dx

.

Using

x \to 2-x

, we get:

I = \int_0^2 \frac{(2-x)^2}{(2-x)^2+x^2} dx = \int_0^2 \frac{(2-x)^2}{x^2+(2-x)^2} dx

.

Adding the two expressions:

2I = \int_0^2 \frac{x^2+(2-x)^2}{x^2+(2-x)^2} dx = \int_0^2 1\,dx = [x]_0^2 = 2

.

Thus

I = 1

.

Answer:

1

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