# 16.9: Derivatives and Primitives (Indefinite Integrals)

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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.