Answer:
0.001591
Step-by-step explanation:
The power series for arctan(x) is:
arctan(x) = ∑ (-1)ⁿ x²ⁿ⁺¹ / (2n + 1)
Substituting 5x:
arctan(5x) = ∑ (-1)ⁿ (5x)²ⁿ⁺¹ / (2n + 1)
Multiply both sides by x:
x arctan(5x) = ∑ (-1)ⁿ x (5x)²ⁿ⁺¹ / (2n + 1)
Simplify:
x arctan(5x) = ∑ (-1)ⁿ (5x) (5x)²ⁿ⁺¹ / (10n + 5)
x arctan(5x) = ∑ (-1)ⁿ (5x)²ⁿ⁺² / (10n + 5)
Multiply top and bottom by 5:
x arctan(5x) = ∑ (-1)ⁿ 5 (5x)²ⁿ⁺² / (50n + 25)
Integrate:
∫ x arctan(5x) = ∑ (-1)ⁿ (5x)²ⁿ⁺³ / ((50n + 25) (2n + 3))
Evaluate between x = 0.1 and x = 0:
∫₀⁰¹ x arctan(5x) = [∑ (-1)ⁿ (5x)²ⁿ⁺³ / ((50n + 25) (2n + 3))] |₀⁰¹
∫₀⁰¹ x arctan(5x) = ∑ (-1)ⁿ (0.5)²ⁿ⁺³ / ((50n + 25) (2n + 3))
This is an alternating series. We can approximate it with Alternating Series Estimation.
bₙ₊₁ ≥ ε
(0.5)²⁽ⁿ⁺¹⁾⁺³ / ((50(n+1) + 25) (2(n+1) + 3)) ≥ 0.000001
(0.5)²ⁿ⁺⁵ / ((50n + 75) (2n + 5)) ≥ 0.000001
n ≥ 3
So the approximation is the sum of the terms from n=0 to n=3.
(-1)⁰ (0.5)²⁽⁰⁾⁺³ / ((50(0) + 25) (2(0) + 3))
+ (-1)¹ (0.5)²⁽¹⁾⁺³ / ((50(1) + 25) (2(1) + 3))
+ (-1)² (0.5)²⁽²⁾⁺³ / ((50(2) + 25) (2(2) + 3))
+ (-1)³ (0.5)²⁽³⁾⁺³ / ((50(3) + 25) (2(3) + 3))
= 0.0016667 − 0.0000833 + 0.0000089 − 0.0000012
= 0.001591
