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Wize University Calculus 1 Textbook > Integrals

Antiderivatives of 1x\frac{1}{x}

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Antiderivatives (Indefinite Integral) of 1x\color{blue}\frac{1}{x}

Let kk be a nonzero constant. Then we have the following indefinite integral rules:
∫1x dx=ln⁡∣x∣+C, (for x≠0)\boxed{\displaystyle\int\dfrac{1}{x} \ dx=\ln|x|+C,\ \text{(for}\ x\neq0)}

∫1kx+adx=1kln⁡∣kx+a∣+C,(for x≠0,a constant)\displaystyle \boxed{\int\frac{1}{kx+a}dx=\frac{1}{k}\ln|kx+a|+C }, \text{(for } x\ne0, a \text{ constant})

Watch Out!
Notice the absolute value bars inside the natural logarithm!

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Example: Indefinite integrals of 1x\color{blue}\frac{1}{x}

Evaluate the following indefinite integral

∫x2+x+2x2dx\displaystyle \int \frac{x^2+x+2}{x^2}dx

∫x2+x+2x2dx\displaystyle \int \frac{x^2+x+2}{x^2}dx

=∫(x2x2+xx2+2x2)dx=\displaystyle \int \bigg(\frac{x^2}{x^2}+\frac{x}{x^2}+\frac{2}{x^2}\bigg)dx

=∫(1+1x+2x−2)dx=\displaystyle \int(1+\frac{1}{x}+2x^{-2})dx

=x+ln⁡∣x∣+2x−1−1+C\displaystyle = x+\ln |x|+\frac{2x^{-1}}{-1}+C

=x+ln⁡∣x∣−2x+C\displaystyle =\boxed{ x+\ln |x|-\frac{2}{x}+C}

Evaluate the following indefinite integral

∫34−2x dx\displaystyle \int \frac{3}{4-2x} \ dx
Extra Practice
Antiderivatives: Indefinite integrals
Evaluate the integral ∫1+1x+2x2+3x3dx\displaystyle\int_{ }^{ }1+\frac{1}{x}+\frac{2}{x^2}+\frac{3}{x^3}dx.
Antiderivatives: Indefinite integrals
Evaluate the integral ∫1+1x+2x2+3x3dx\displaystyle\int_{ }^{ }1+\frac{1}{x}+\frac{2}{x^2}+\frac{3}{x^3}dx.
Indefinite Integral
∫(x−2+x−1+1+x+x2)dx\displaystyle \int_{ }^{ }\left(x^{-2}+x^{-1}+1+x+x^2\right)dx
Practice: Long Division First
Q:\bf{Q:} Evaluate ∫x3−x2−x−3x−2dx\displaystyle\int_{ }^{ }\frac{x^3-x^2-x-3}{x-2}dx
Practice: Long Division First
Evaluate ∫x3−x2−x−3x−2dx\displaystyle\int_{ }^{ }\frac{x^3-x^2-x-3}{x-2}dx
(a) x33+x22+x−ln⁡∣x−2∣+C\frac{x^3}{3}+\frac{x^2}{2}+x-\ln\left|x-2\right|+C (b) 12x2−3ln⁡∣x−2∣+C\frac12x^2-3\ln\left|x-2\right|+C (c) =x33+x22+x−12(x−2)2+C=\frac{x^3}{3}+\frac{x^2}{2}+x-\frac12(x-2)^2+C
Practice: Indefinite Integral
Evaluate the integral ∫x+1+1x+2x2dx\displaystyle\int_{ }^{ }x+1+\frac{1}{x}+\frac{2}{x^2}dx
Practice: Indefinite Integral
Evaluate the integral ∫1+1x+2x2+3x3dx\displaystyle\int_{ }^{ }1+\frac{1}{x}+\frac{2}{x^2}+\frac{3}{x^3}dx.
Antiderivatives
Evaluate the following indefinite integral
∫34−2x dx\displaystyle \int \frac{3}{4-2x} \ dx
Practice: Long Division First
Q:\bf{Q:} Evaluate ∫x3−x2−x−3x−2dx\displaystyle\int_{ }^{ }\frac{x^3-x^2-x-3}{x-2}dx
Antiderivatives: Indefinite integrals
Evaluate the integral ∫1+1x+2x2+3x3dx\displaystyle\int_{ }^{ }1+\frac{1}{x}+\frac{2}{x^2}+\frac{3}{x^3}dx.
Indefinite Integral
∫(x−2+x−1+1+x+x2)dx\displaystyle \int_{ }^{ }\left(x^{-2}+x^{-1}+1+x+x^2\right)dx