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HomeLớp 12Toán lớp 12 - Sách Kết nối tri thứcBài tập cuối chương 4 - NGUYÊN HÀM VÀ TÍCH PHÂNBài tập trung bình
{"common":{"save":0,"post_id":"8205","level":2,"total":10,"point":10,"point_extra":0},"segment":[{"id":"6003","post_id":"8205","mon_id":"1159285","chapter_id":"1159380","question":"<p>Cho F(x) là m\u1ed9t nguyên hàm c\u1ee7a hàm s\u1ed1 <span class=\"math-tex\">$f(x)=e^{x^2}\\big(x^3-4x\\big)$<\/span>. Hàm s\u1ed1 F(x) có bao nhiêu \u0111i\u1ec3m c\u1ef1c tr\u1ecb?<\/p>","options":["<strong>A.<\/strong> 1","<strong>B.<\/strong> 2","<strong>C.<\/strong> 3","<strong>D.<\/strong> 0"],"correct":"3","level":"2","hint":"<p>S\u1eed d\u1ee5ng \u0111\u1ecbnh ngh\u0129a nguyên hàm<\/p><p>Cho hàm s\u1ed1 f(x) xác \u0111\u1ecbnh trên K.<\/p><p>Hàm s\u1ed1 F(x) \u0111\u01b0\u1ee3c g\u1ecdi là nguyên hàm c\u1ee7a hàm s\u1ed1 f(x) n\u1ebfu <span class=\"math-tex\">$F^\\prime(x)=f(x)$<\/span>, v\u1edbi m\u1ecdi <span class=\"math-tex\">$x\\in K$<\/span>.<\/p><p><span class=\"math-tex\">$F^\\prime(x)=f(x)$<\/span> ⇔ <span class=\"math-tex\">$F^\\prime(x)=e^{x^2}\\big(x^3-4x\\big)$<\/span>. Tìm c\u1ef1c tr\u1ecb c\u1ee7a hàm s\u1ed1 F(x).<\/p>","answer":"<p>Ch\u1ecdn <span style=\"color:#16a085;\"><strong>C.<\/strong> 3<\/span><\/p><p>Ta có: <span class=\"math-tex\">$F(x)=\\int f(x)dx$<\/span> ⇔ <span class=\"math-tex\">$F^\\prime(x)=f(x)$<\/span> ⇔ <span class=\"math-tex\">$F^\\prime(x)=e^{x^2}\\big(x^3-4x\\big)$<\/span><\/p><p><span class=\"math-tex\">$F^\\prime(x)=0$<\/span> ⇔ <span class=\"math-tex\">$e^{x^2}\\big(x^3-4x\\big)=0$<\/span> ⇔ x = 0 ho\u1eb7c x = 1 ho\u1eb7c x = –1.<\/p><p>Ta th\u1ea5y ph\u01b0\u01a1ng trình <span class=\"math-tex\">$F^\\prime(x)=0$<\/span> có 3 nghi\u1ec7m phân bi\u1ec7t nên <span class=\"math-tex\">$F^\\prime(x)$<\/span> \u0111\u1ed5i d\u1ea5u qua c\u1ea3 3 nghi\u1ec7m \u0111ó.<\/p><p>V\u1eady F(x) có 3 \u0111i\u1ec3m c\u1ef1c tr\u1ecb.<\/p>","type":"choose","extra_type":"shape1","user_id":"131","test":"0","date":"2024-12-22 11:53:40","option_type":"txt","len":0},{"id":"6006","post_id":"8205","mon_id":"1159285","chapter_id":"1159380","question":"<p>Cho hàm s\u1ed1 <span class=\"math-tex\">$f(x)=\\dfrac{2x+1}{2x-3}$<\/span> xác \u0111\u1ecbnh trên <span class=\"math-tex\">$\\bigg(\\dfrac{3}{2};+\\infty\\bigg)$<\/span>; F(x) là m\u1ed9t nguyên hàm c\u1ee7a hàm s\u1ed1 <span class=\"math-tex\">$f(x)=\\dfrac{2x+1}{2x-3}$<\/span> th\u1ecfa mãn <span class=\"math-tex\">$F(2) = 3$<\/span>. Tìm <span class=\"math-tex\">$F(x)$<\/span>?<\/p>","options":["<strong>A.<\/strong> <span class=\"math-tex\">$F(x)=x+4\\ln(2x-3)+1$<\/span>","<strong>B.<\/strong> <span class=\"math-tex\">$F(x)=x+\\ln(2x-3)+1$<\/span>","<strong>C.<\/strong> <span class=\"math-tex\">$F(x)=x+2\\ln(2x-3)+C$<\/span>","<strong>D.<\/strong> <span class=\"math-tex\">$F(x)=x+2\\ln(2x-3)+1$<\/span>"],"correct":"4","level":"2","hint":"<p>S\u1eed d\u1ee5ng công th\u1ee9c nguyên hàm <\/p><p><span class=\"math-tex\">$\\int dx=x+C$<\/span><\/p><p><span class=\"math-tex\">$\\int \\dfrac{1}{ax+b}dx=\\dfrac{1}{a}\\ln|ax+b|+C$<\/span><\/p>","answer":"<p>Ch\u1ecdn <span style=\"color:#16a085;\"><strong>D.<\/strong> <span class=\"math-tex\">$F(x)=x+2\\ln(2x-3)+1$<\/span><\/span><\/p><p><span class=\"math-tex\">$F(x)=\\int \\dfrac{2x+1}{2x-3}dx=\\int\\bigg(1+\\dfrac{4}{2x-3} \\bigg)dx$<\/span> <span class=\"math-tex\">$=x+2\\ln|2x-3|+C$<\/span> <span class=\"math-tex\">$=x+2\\ln(2x-3)+C$<\/span><\/p><p>Mà <span class=\"math-tex\">$F(2) = 3$<\/span> ⇔ <span class=\"math-tex\">$2+2\\ln(2.2-3)+C=3$<\/span> ⇔ C = 1.<\/p><p>V\u1eady <span class=\"math-tex\">$F(x)=x+2\\ln(2x-3)+1$<\/span>.<\/p>","type":"choose","extra_type":"classic","user_id":"131","test":"0","date":"2024-12-23 08:07:59","option_type":"math","len":0},{"id":"6007","post_id":"8205","mon_id":"1159285","chapter_id":"1159380","question":"<p>Cho hàm s\u1ed1 <span class=\"math-tex\">$f(x)$<\/span> xác \u0111\u1ecbnh trên \u211d\\{1} th\u1ecfa mãn <span class=\"math-tex\">$f^\\prime(x)=\\dfrac{1}{x-1}$<\/span> và <span class=\"math-tex\">$f(0)=2019$<\/span>; <span class=\"math-tex\">$f(2)=2020$<\/span>. Tính <span class=\"math-tex\">$S=f(3)-f(-1)$<\/span>.<\/p>","options":["<strong>A.<\/strong> <span class=\"math-tex\">$2\\ln2+4039$<\/span>","<strong>B.<\/strong> 4039","<strong>C.<\/strong> 1","<strong>D.<\/strong> –1"],"correct":"3","level":"2","hint":"<p>S\u1eed d\u1ee5ng công th\u1ee9c nguyên hàm <\/p><p><span class=\"math-tex\">$\\int dx=x+C$<\/span><\/p><p><span class=\"math-tex\">$\\int \\dfrac{1}{ax+b}dx=\\dfrac{1}{a}\\ln|ax+b|+C$<\/span><\/p>","answer":"<p>Ch\u1ecdn <span style=\"color:#16a085;\"><strong>C.<\/strong> 1<\/span><\/p><p><span class=\"math-tex\">$f(x)=\\int f^\\prime(x)dx=\\int\\dfrac{1}{x-1}dx=\\ln|x-1|+C$<\/span> <span class=\"math-tex\">$=\\begin{cases}\\ln(x-1)+C_1,x>1\\\\\\ln(1-x)+C_2,x<1\\end{cases}$<\/span><\/p><p><span class=\"math-tex\">$f(0)=2019$<\/span> ⇒ <span class=\"math-tex\">$C_2=2019$<\/span><\/p><p><span class=\"math-tex\">$f(2)=2020$<\/span> ⇒ <span class=\"math-tex\">$C_1=2020$<\/span><\/p><p>Suy ra <span class=\"math-tex\">$f(x)=\\begin{cases}\\ln(x-1)+2020,x>1\\\\\\ln(1-x)+2019,x<1\\end{cases}$<\/span><\/p><p><span class=\"math-tex\">$S=f(3)-f(-1)=\\ln2+2020-\\ln2-2019=1$<\/span><\/p>","type":"choose","extra_type":"classic","user_id":"131","test":"0","date":"2024-12-23 08:12:08","option_type":"math","len":0}]}