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16.9: Derivatives and Primitives (Indefinite Integrals)

  • Page ID
    107068
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    \(f(x)\) \(f'(x)\) \(\int f(x)dx\)(\(\pm c\))
    \(k\) 0 \(kx\)
    \(x^n\) \(n x^{n-1}\), \(n \neq 0\) \(\frac{x^{n+1}}{n+1}\), \(n \neq -1\)
    \(\frac{1}{x}\) \(-\frac{1}{x^2}\) \(ln|x|\)
    \(a^x\) \(a^x \ln{a}\) \(\frac{a^x}{\ln{a}}\)
    \(e^x\) \(e^x\) \(e^x\)
    \(\log_a{x}\) \(\frac{1}{x \ln{a}}\) \(\frac{x \ln{x}-x}{\ln{a}}\)
    \(\ln{x}\) \(\frac{1}{x}\) \(x \ln{x}-x\)
    \(\sin{x}\) \(\cos{x}\) \(-\cos{x}\)
    \(\cos{x}\) \(-\sin{x}\) \(\sin{x}\)
    \(\tan{x}\) \(\frac{1}{\cos^2{x}}\) \(-\ln{(\cos{x}})\)
    \(\arcsin{x}\) \(\frac{1}{\sqrt{1-x^2}}\) \(x \arcsin{x}+\sqrt{1-x^2}\)
    \(\arccos{x}\) \(-\frac{1}{\sqrt{1-x^2}}\) \(x \arccos{x}-\sqrt{1-x^2}\)
    \(\arctan{x}\) \(\frac{1}{1+x^2}\) \(x \arctan{x}-\frac{1}{2}\ln{(1+x^2)}\)

    \(\frac{1}{a^2+x^2}\)

    \(\frac{-2x}{(a^2+x^2)^2}\) \(\frac{1}{a}\arctan{\left( \frac{x}{a}\right)}\)

    \(\frac{1}{\sqrt{a^2-x^2}}\)

    \(\frac{x}{(a^2-x^2)^{\frac{3}{2}}}\) \(\arcsin{\left( \frac{x}{a}\right)}\)
    • \(\int \sin^2{(ax)}dx= \frac{x}{2}-\frac{\sin{(2ax)}}{4a}+c\)
    • \(\int \cos^2{(ax)}dx= \frac{x}{2}+\frac{\sin{(2ax)}}{4a}+c\)
    • \(\int \sin^3{(ax)}dx= \frac{1}{12a}\cos{(3ax)}-\frac{3}{4a}\cos{(ax)}+c\)
    • \(\int \cos^3{(ax)}dx= \frac{1}{12a}\sin{(3ax)}+\frac{3}{4a}\sin{(ax)}+c\)
    • \(\int x \cos{(ax)}dx=\frac{\cos{(ax)}}{a^2}+\frac{\sin{(ax)}}{a}x+c\)
    • \(\int x \sin{(ax)}dx=\frac{\sin{(ax)}}{a^2}-\frac{\cos{(ax)}}{a}x+c\)
    • \(\int x \sin^2{(ax)}dx=\frac{x^2}{4}-\frac{x\sin{(2ax)}}{4a}-\frac{\cos{(2ax)}}{8a^2}+c\)
    • \(\int xe^{x^2}dx=e^{x^{2}}/2+c\)
    • \(\int x e^{ax}=\frac{e^{ax}(ax-1)}{a^2}+c\)
    • \(\int \frac{x}{x^2+1}dx=\frac{1}{2}\ln{(1+x^2)} +c\)

    This page titled 16.9: Derivatives and Primitives (Indefinite Integrals) is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Marcia Levitus via source content that was edited to the style and standards of the LibreTexts platform; a detailed edit history is available upon request.

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