Mastering basic integration formulas is an essential step in learning calculus. These formulas provide quick methods for finding antiderivatives of common algebraic, exponential, logarithmic, and trigonometric functions. By understanding and memorizing these standard integrals, students can solve a wide range of integration problems more efficiently and build a strong foundation for advanced topics such as substitution, integration by parts, and differential equations. The table below summarizes the most important indefinite integral formulas frequently used in mathematics courses and examinations.
- $\int dx = x + C$
- $\int x^\alpha dx = \dfrac{x^{\alpha+1}}{\alpha+1} + C \qquad (\alpha \ne -1)$
- $\int \dfrac{dx}{x} = \ln|x| + C \qquad (x \ne 0)$
- $\int e^x dx = e^x + C$
- $\int a^x dx = \dfrac{a^x}{\ln a} + C \qquad (0 < a \ne 1)$
- $\int \cos x , dx = \sin x + C$
- $\int \sin x , dx = -\cos x + C$
- $\int \dfrac{dx}{\cos^2 x} = \int (1+\mathrm{tg}^2x)dx = \mathrm{tg}x + C$
- $\int \dfrac{dx}{\sin^2 x} = \int (1+\mathrm{cotg}^2x)dx = -\mathrm{cotg}x + C$
- $\int \dfrac{dx}{2\sqrt{x}} = \sqrt{x} + C \qquad (x > 0)$
- $\int du = u + C$
- $\int u^\alpha du = \dfrac{u^{\alpha+1}}{\alpha+1} + C \qquad (\alpha \ne -1)$
- $\int \dfrac{du}{u} = \ln|u| + C \qquad (u = u(x) \ne 0)$
- $\int e^u du = e^u + C$
- $\int a^u du = \dfrac{a^u}{\ln a} + C \qquad (0 < a \ne 1)$
- $\int \cos u , du = \sin u + C$
- $\int \sin u , du = -\cos u + C$
- $\int \dfrac{du}{\cos^2 u} = \int (1+\mathrm{tg}^2u)du = \mathrm{tg}u + C$
- $\int \dfrac{du}{\sin^2 u} = \int (1+\mathrm{cotg}^2u)du = -\mathrm{cotg},u + C$
- $\int \dfrac{du}{2\sqrt{u}} = \sqrt{u} + C \qquad (u > 0)$
- $\int \cos(ax+b),dx = \dfrac{1}{a}\sin(ax+b) + C \qquad (a \ne 0)$
- $\int \sin(ax+b),dx = -\dfrac{1}{a}\cos(ax+b) + C \qquad (a \ne 0)$
- $\int \dfrac{dx}{ax+b} = \dfrac{1}{a}\ln|ax+b| + C$
- $\int e^{ax+b}dx = \dfrac{1}{a}e^{ax+b} + C \qquad (a \ne 0)$
- $\int \dfrac{dx}{\sqrt{ax+b}} = \dfrac{2}{a}\sqrt{ax+b} + C \qquad (a \ne 0)$
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