Consider the rational function
𝑓
(
𝑥
)
=
(5
𝑥
+2)/(
𝑥
2
+1)

Find
∫
0
1
𝑓
(
𝑥
)
𝑑
𝑥

Enter your answer to 4 decimal places.

Question

Consider the rational function 
𝑓
(
𝑥
)
=
(5
𝑥
+2)/(
𝑥
2
+1)

Find 
∫
0
1
𝑓
(
𝑥
)
𝑑
𝑥

Enter your answer to 4 decimal places.

BlackTom AI · Accepted Answer

Numerator 5x+2 can be split to match derivatives of x^2+1 and a constant term: (5x+2)/(x^2+1) = (5/2)*(2x)/(x^2+1) + 2/(x^2+1). The relevant knowledge points are: ∫(2x)/(x^2+1) dx = ln(x^2+1) (chain rule) and ∫1/(x^2+1) dx = arctan x.

Step-by-step work:
1. Split the integral:
   ∫_0^1 (5x+2)/(x^2+1) dx = (5/2) ∫_0^1 (2x)/(x^2+1) dx + 2 ∫_0^1 1/(x^2+1) dx.

2. Integrate each term using the known antiderivatives:
   (5/2) ∫ (2x)/(x^2+1) dx = (5/2) ln(x^2+1),
   2 ∫ 1/(x^2+1) dx = 2 arctan x.

3. Evaluate from 0 to 1:
   (5/2)[ln(x^2+1)]_0^1 + 2[arctan x]_0^1
   = (5/2)(ln(1^2+1) - ln(0^2+1)) + 2(arctan 1 - arctan 0)
   = (5/2) ln 2 + 2*(π/4)
   = (5/2) ln 2 + π/2.

4. Numeric calculation (showing intermediate steps):
   ln 2 ≈ 0.69314718056 → (5/2) ln 2 = 2.5 * 0.69314718056 = 1.7328679514.
   π/2 ≈ 1.57079632679.
   Sum = 1.7328679514 + 1.57079632679 = 3.30366427819.

Rounding to four decimal places gives 3.3037.

Why this is correct: the decomposition uses the derivative of the denominator for the x-term and the standard arctan integral for the constant term; both antiderivatives are exact and were evaluated correctly at the bounds. Common incorrect approaches include forgetting the 1/2 factor when rewriting 5x as (5/2)*2x, or failing to multiply arctan(1)=π/4 by 2 — these produce wrong decimals. No other answer choices apply here.

Final answer (to four decimal places): 3.3037, which matches the provided answer.

Consider the rational function 𝑓 ( 𝑥 ) = (5 𝑥 +2)/( 𝑥 2 +1) Find ∫ 0 1 𝑓 ( 𝑥 ) 𝑑 𝑥 Enter your answer to 4 decimal places.简答题

类似问题

Find the value of [math: ∫04(−x+3x)dx]

The exact value of [math: ∫−22(4e2x−2e−2x)dx] is:

以下哪一项等于\[\int _{-\pi }^{\pi } f（x） dx\]，正确到小数3位，其中 \[f（x） = \begin{cases} - 7 x & \text {if } -\pi \leq x \leq 0 \\ 3 - 3 \cos {x} & \text {if } 0 < x \leq \pi \end{cases} 。\]

以下哪一项等于 ∫ π −π f（x）dx ，正确到小数3位，其中 f（x）={−7x 如果−π≤x≤0 3⁻³因为x 如果0<x≤π。

Which of the following is equal to ∫ − 𝜋 𝜋 𝑓 ( 𝑥 ) 𝑑 𝑥 correct to 3 decimal places, where 𝑓 ( 𝑥 ) = { 12 𝑥 if − 𝜋 ≤ 𝑥 ≤ 0 3 − 3 cos ⁡ 𝑥 if 0 < 𝑥 ≤ 𝜋 .

Evaluate .

The velocity 𝑣 of a particle at time 𝑡 (in seconds) is measured and the graph 𝑦 = 𝑣 ( 𝑡 ) is shown below. What is the particle's displacement after the first 6 seconds?

更多留学生实用工具

智能学习助手

风格化写作助手

论文查重助手

文献引用助手

课堂转译助手

课堂笔记助手

Quiz搜索助手

学校历年真题

智能刷题助手

智能匹配练习

希望你的学习变得更简单