Senin, 23 September 2013

Integral Tak tentu

Kesulitan menghadapi banyak sekali jenis integral? Kali ini penulis akan memaparkan beberapa jenis integral tak tentu yang umum dijumpai dalam berbagai soal matematika. Bentuk-bentuk berikut ini merupakan bentuk dari yang paling dasar sampai dengan bentuk pengembangan.
Integral tak tentu untuk fungsi aljabar (dasar)
  1. \displaystyle\int dx = x + C
  2. \displaystyle\int x^n \,dx = \frac{1}{n+1}x^{n+1}+C,\qquad n\ne-1
  3. \displaystyle\int ax^n \,dx = \frac{a}{n+1}x^{n+1}+C,\qquad n\ne-1
  4. \displaystyle\int (ax+b)^n \,dx= \frac{1}{a(n+1)}(ax+b)^{n+1}+C,\qquad n\ne-1
Integral tak tentu untuk fungsi trigonometri (dasar)
  1. \displaystyle\int\cos{x}\,dx=\sin{x}+C
  2. \displaystyle\int\sin{x}\,dx=-\cos{x}+C
  3. \displaystyle\int\sec^2{x}\,dx=\tan{x}+C
  4. \displaystyle\int\csc^2{x}\,dx=-\cot{x}+C
  5. \displaystyle\int\tan{x}\sec{x}\,dx=\sec{x}+C
  6. \displaystyle\int\cot{x}\csc{x}\,dx=-\csc{x}+C
Integral tak tentu untuk fungsi trigonometri (2) untuk a\ne0
  1. \displaystyle\int\cos(ax+b)\,dx=\dfrac{\sin(ax+b)}{a}+C
  2. \displaystyle\int\sin(ax+b)\,dx=-\dfrac{\cos(ax+b)}{a}+C
  3. \displaystyle\int\sec^2(ax+b)\,dx=\dfrac{\tan(ax+b)}{a}+C
  4. \displaystyle\int\csc^2(ax+b)\,dx=-\dfrac{\cot(ax+b)}{a}+C
  5. \displaystyle\int\tan(ax+b)\sec(ax+b)\,dx=\dfrac{\sec(ax+b)}{a}+C
  6. \displaystyle\int\cot(ax+b)\csc(ax+b)\,dx=-\dfrac{\csc(ax+b)}{a}+C
Integral tak tentu untuk fungsi trigonometri (3)
  1. \displaystyle\int\cos^2{x}\,dx=\dfrac{x}{2}+\dfrac{\sin{2x}}{4}+C
  2. \displaystyle\int\sin^2{x}\,dx=\dfrac{x}{2}-\dfrac{\sin{2x}}{4}+C
  3. \displaystyle\int\tan^2{x}\,dx=\tan{x}-x+C
  4. \displaystyle\int\cot^2{x}\,dx=\cot{x}-x+C
  5. \displaystyle\int\cos^3{x}\,dx=\dfrac{1}{3}(2+\cos^2{x})\sin{x}+C*
  6. \displaystyle\int\sin^3{x}\,dx=-\dfrac{1}{3}(2+\sin^2{x})\cos{x}+C*
*ada kemungkinan bentuk yang lain dengan menjabarkan cos x
Integral tak tentu untuk fungsi trigonometri (4) -hard level-
  1. \displaystyle\int\sin{ax}\sin{bx}\,dx = \dfrac{\sin{(a-b)x}}{2(a-b)}-\dfrac{\sin{(a+b)x}}{2(a+b)}+C
  2. \displaystyle\int\cos{ax}\cos{bx}\,dx = \dfrac{\sin{(a-b)x}}{2(a-b)}+\dfrac{\sin{(a+b)x}}{2(a+b)}+C
  3. \displaystyle\int\sin{ax}\cos{bx}\,dx = -\dfrac{\cos{(a-b)x}}{2(a-b)}-\dfrac{\cos{(a+b)x}}{2(a+b)}+C
  4. \displaystyle\int\sin^n{x}\,dx=-\dfrac{sin^{n-1}{x}\cos{x}}{n}+\dfrac{n-1}{n}\int{\sin^{n-2}x}\,dx
  5. \displaystyle\int\cos^n{x}\,dx=\dfrac{cos^{n-1}{x}\sin{x}}{n}+\dfrac{n-1}{n}\int{\cos^{n-2}x}\,dx
  6. \displaystyle\int\tan^n{x}\,dx=\dfrac{\tan^{n-1}x}{n-1}-\int{\tan^{n-2}x}\,dx
  7. \displaystyle\int\cot^n{x}\,dx=-\dfrac{\cot^{n-1}x}{n-1}-\int{\cot^{n-2}x}\,dx
Integral tak tentu untuk fungsi aljabar (2)
  1. \displaystyle\int\sqrt{a^2-x^2}\,dx=\dfrac{x}{2}\sqrt{a^2-x^2}+a^2\arcsin{\left (\dfrac{x}{a}\right )}+C
  2. \displaystyle\int \dfrac{1}{\sqrt{a^2-x^2}}\,dx=\dfrac{1}{a}\arcsin{\left (\dfrac{x}{a}\right )}+C
  3. \displaystyle\int \dfrac{1}{a^2+x^2}\,dx=\dfrac{1}{a}\arctan{\left (\dfrac{x}{a}\right )}
Integral tak tentu untuk fungsi aljabar (Pengayaan)
  1. \displaystyle\int\dfrac{1}{x}\,dx=\ln{|x|}+C
  2. \displaystyle\int\dfrac{c}{ax+b}\,dx=\dfrac{c}{a}\ln{|ax+b|}+C
  3. \displaystyle\int\dfrac{1}{a^2-x^2}\,dx=\dfrac{1}{2a}\ln{\left |\dfrac{x+a}{x-a}\right |}+C
  4. \displaystyle\int \dfrac{1}{x\sqrt{x^2-a^2}}\,dx=\dfrac{1}{a}\sec^{-1}{\left |\dfrac{x}{a}\right |}+C
Integral tak tentu untuk fungsi eksponen (Pengayaan)
  1. \displaystyle\int e^x dx= e^x+C
  2. \displaystyle\int e^{ax} dx= \dfrac{1}{a}e^{ax}+C
  3. \displaystyle\int a^x dx= \dfrac{a^x}{\ln{a}}+C
  4. \displaystyle\int a^{kx} dx= \dfrac{a^{kx}}{k\ln{a}}+C
Integral tak tentu untuk fungsi logritma (Pengayaan)
  1. \displaystyle\int\ln{x}\,dx=x\cdot\ln{x}-x+C
  2. \displaystyle\int\,^a\ln{x}\,dx=x\cdot^a\ln{x}-\dfrac{x}{\ln{a}}+C
Integral tak tentu untuk fungsi trigonometri (Pengayaan)
  1. \displaystyle\int\tan{x}\,dx=\ln{|\sec{x}|}+C
  2. \displaystyle\int\cot{x}\,dx=\ln{|\sin{x}|}+C
  3. \displaystyle\int\sec{x}\,dx=\ln{|\sec{x}+\tan{x}|}+C
  4. \displaystyle\int\csc{x}\,dx=\ln{|\csc{x}+\cot{x}|}+C
  5. \displaystyle\int\tan^3{x}\,dx=\dfrac{1}{2}\tan^2{x}+\ln{|\sec{x}|}+C
  6. \displaystyle\int\cot^3{x}\,dx=-\dfrac{1}{2}\cot^2{x}-\ln{|\sin{x}|}+C
  7. \displaystyle\int\sec^3{x}\,dx=\dfrac{1}{2}\sec{x}\tan{x}+\ln{|\sec{x}+\tan{x}|}+C
  8. \displaystyle\int\csc^3{x}\,dx=-\dfrac{1}{2}\csc{x}\cot{x}+\ln{|\csc{x} -\cot{x}|}+C
Catatan:
Bentuk-bentuk di atas kebanyakan merupakan pengembangan dari bentuk-bentuk dasar dan perpaduan antara dua/lebih jenis integral
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