Evaluate the definite integral. Enter exact values.
[tex]\int_0^1-\frac{2 \cot ^{-1}(x)}{1+x^2} d x=[/tex]
Answer (2)
Substitute u = cot − 1 ( x ) , so d u = − 1 + x 2 1 d x .
Change the limits of integration: when x = 0 , u = 2 π , and when x = 1 , u = 4 π .
Evaluate the integral: ∫ 2 π 4 π 2 u d u = [ u 2 ] 2 π 4 π = ( 4 π ) 2 − ( 2 π ) 2 .
Simplify to find the final answer: − 16 3 π 2 .
Explanation
Problem Analysis and Setup We are asked to evaluate the definite integral ∫ 0 1 − 1 + x 2 2 cot − 1 ( x ) d x We will use u-substitution to solve this problem.
U-Substitution Let u = cot − 1 ( x ) . Then, the derivative of u with respect to x is d x d u = − 1 + x 2 1 . Therefore, d u = − 1 + x 2 1 d x . We can rewrite the integral in terms of u .
Evaluating the Integral When x = 0 , u = cot − 1 ( 0 ) = 2 π . When x = 1 , u = cot − 1 ( 1 ) = 4 π . The integral becomes ∫ 2 π 4 π − 2 u ( − d u ) = ∫ 2 π 4 π 2 u d u = [ u 2 ] 2 π 4 π = ( 4 π ) 2 − ( 2 π ) 2 = 16 π 2 − 4 π 2 = 16 π 2 − 16 4 π 2 = − 16 3 π 2
Final Answer Thus, the value of the definite integral is − 16 3 π 2
Examples
Imagine you're calculating the total change in angular momentum of a spinning object as its rate of spin varies. The integral you solved is analogous to finding the accumulated change, where the inverse cotangent function might describe how the object's resistance to spinning changes over time. By evaluating this integral, you determine the net effect on the object's rotational behavior, crucial for designing stable and efficient rotating machinery.
To evaluate the integral ∫ 0 1 − 1 + x 2 2 c o t − 1 ( x ) d x , we perform a u-substitution with u = cot − 1 ( x ) which changes the limits of integration. After evaluating the resulting integral, we find that the final answer is − 16 3 π 2 .
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