Biết .<span class="MathJax_Preview" style="color: inherit;"><span class="MJXp-math" id="MJXp-Span-1"><span class="MJXp-mstyle" id="MJXp-Span-2"><span class="MJXp-msubsup" id="MJXp-Span-3"><span class="MJXp-mo" id="MJXp-Span-4" style="margin-left: 0em; margin-right: 0.05em;">∫</span><span class="MJXp-script-box" style="height: 1.86em; vertical-align: -0.64em;"><span class=" MJXp-script"><span><span style="margin-bottom: -0.25em;"><span class="MJXp-mi MJXp-italic" id="MJXp-Span-6">π</span></span></span></span><span class=" MJXp-script"><span><span style="margin-top: -0.85em;"><span class="MJXp-mn" id="MJXp-Span-5">0</span></span></span></span></span></span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-7">f</span><span class="MJXp-mo" id="MJXp-Span-8" style="margin-left: 0em; margin-right: 0em;">(</span><span class="MJXp-mi" id="MJXp-Span-9">sin</span><span class="MJXp-mo" id="MJXp-Span-10" style="margin-left: 0em; margin-right: 0em;"></span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-11">x</span><span class="MJXp-mo" id="MJXp-Span-12" style="margin-left: 0em; margin-right: 0em;">)</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-13">d</span><span class="MJXp-mi MJXp-italic" id="MJXp-Span-14">x</span><span class="MJXp-mo" id="MJXp-Span-15" style="margin-left: 0.333em; margin-right: 0.333em;">=</span><span class="MJXp-mn" id="MJXp-Span-16">1</span></span></span></span><span id="MathJax-Element-1-Frame" class="mjx-chtml MathJax_CHTML MJXc-processed" tabindex="0" style="font-size: 127%;"><span id="MJXc-Node-1" class="mjx-math"><span id="MJXc-Node-2" class="mjx-mrow"><span id="MJXc-Node-3" class="mjx-mstyle"><span id="MJXc-Node-4" class="mjx-mrow" style="font-size: 120%;"><span id="MJXc-Node-5" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.138em;"><span id="MJXc-Node-6" class="mjx-mo" style="padding-right: 0.138em;"><span class="mjx-char MJXc-TeX-size1-R" style="padding-top: 0.593em; padding-bottom: 0.593em;">∫</span></span></span><span class="mjx-stack" style="vertical-align: -0.366em;"><span class="mjx-sup" style="font-size: 70.7%; padding-bottom: 0.579em; padding-left: 0.324em; padding-right: 0.071em;"><span id="MJXc-Node-8" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.199em; padding-bottom: 0.298em; padding-right: 0.003em;">π</span></span></span><span class="mjx-sub" style="font-size: 70.7%; padding-right: 0.071em;"><span id="MJXc-Node-7" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.396em; padding-bottom: 0.347em;">0</span></span></span></span></span><span id="MJXc-Node-9" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.495em; padding-bottom: 0.495em; padding-right: 0.06em;">f</span></span><span id="MJXc-Node-10" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.593em;">(</span></span><span id="MJXc-Node-11" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.396em; padding-bottom: 0.347em;">sin</span></span><span id="MJXc-Node-12" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-13" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.199em; padding-bottom: 0.298em;">x</span></span><span id="MJXc-Node-14" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.445em; padding-bottom: 0.593em;">)</span></span><span id="MJXc-Node-15" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.495em; padding-bottom: 0.298em; padding-right: 0.003em;">d</span></span><span id="MJXc-Node-16" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.199em; padding-bottom: 0.298em;">x</span></span><span id="MJXc-Node-17" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.101em; padding-bottom: 0.298em;">=</span></span><span id="MJXc-Node-18" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.396em; padding-bottom: 0.347em;">1</span></span></span></span></span></span></span><script type="math/tex" id="MathJax-Element-1">\large \int_0^\pi f(\sin x) dx= 1 </script>. Tính $\large \int_0^\pi xf

Biết .π0f(sinx)dx=1π0f(sinx)dx=1. Tính $\large \int_0^\pi xf

4.2/5

Tác giả: Thầy Tùng

Đăng ngày: 18 Aug 2022

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Câu hỏi:

Biết .π0f(sinx)dx=1π0f(sinx)dx=1. Tính π0xf(sinx)dxπ0xf(sinx)dx

Đáp án án đúng là: C

Lời giải chi tiết:

Chọn C

Đặt x=πtdx=dtx=πtdx=dt. Đổi cận x=0t=π;x=πt=0x=0t=π;x=πt=0

Khi đó: π0xf(sinx)dx=0π(πt)f(sin(πt))(dt)π0xf(sinx)dx=0π(πt)f(sin(πt))(dt)

=π0(πt)f(sint)dt=ππ0f(sinx)dxπ0xf(sinx)dx=π0(πt)f(sint)dt=ππ0f(sinx)dxπ0xf(sinx)dx

Do đó: 2π0xf(sinx)dx=ππ0f(sinx)dx=π2π0xf(sinx)dx=ππ0f(sinx)dx=π

Vậy π0xf(sinx)dx=π2π0xf(sinx)dx=π2