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Practice: Indefinite Integral Evaluate the integral ∫_ ^ 1+1/x+2/(x^2)+3/(x^3)d…

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Wize University Calculus 1 Textbook > Integrals

Antiderivatives & The Indefinite Integrals

5 Activities

Wize University Calculus 1 Textbook > Integrals

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

3 Activities

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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.
More Antiderivatives & The Indefinite Integrals Questions:
Practice: Definite Integral
Practice: Definite Integral
Evaluate ∫−20∣x2−1∣dx\displaystyle \int_{-2}^0\left|x^2-1\right|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.
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
If F(x)=ln⁡(2x2)F(x) = \ln(2x^2) is an antiderivative of f(x)f(x), find ff.
Practice: Definite Integral
Practice: Definite Integral
Evaluate ∫−20∣x2−1∣dx\displaystyle \int_{-2}^0\left|x^2-1\right|dx.
Find ∫(x23−x5+1x34)dx\displaystyle \int \left(\sqrt[3]{x^2}-\sqrt[5]{x}+\frac{1}{\sqrt[4]{x^3}}\right)dx
Practice: Indefinite Simplify First
Q:\textbf{Q:} Find ∫[x2−4xx+x(2x2−3x)]dx\displaystyle\int_{ }^{ }\left[\frac{x^2-4\sqrt{x}}{\sqrt{x}}+x\left(2x^2-3x\right)\right]dx
Practice: Indefinite Integral
Evaluate the integral ∫x+1+1x+2x2dx\displaystyle\int_{ }^{ }x+1+\frac{1}{x}+\frac{2}{x^2}dx
Antiderivatives: Exponential Functions
Which of the following is the most general antiderivative of the function e3x+7e^{3x+7}? In the functions below, cc is an arbitrary constant.
Antiderivatives and Indefinite Integrals
Find the most general antiderivative of f(x)=e2f\left(x\right)=e^2
Riemann Sums
Evaluate lim⁡n→∞ ∑i=1n[5n[(1+5in)2−3]]\displaystyle \lim_{n\to\infty}\ \sum_{i=1}^n\left[\frac{5}{n}\left[\left(1+\frac{5i}{n}\right)^2-3\right]\right]
Find ∫xe3 dx\int_{ }^{ }xe^3\ dx
Antiderivatives: Indefinite Integrals
If F(x)=ln⁡(2x2)F(x) = \ln(2x^2) is an antiderivative of f(x)f(x), find ff.
Antiderivatives: Indefinite Integrals
The graph of an antiderivative of f(x)f(x) is given below. Graph f(x)f(x).
Find an antiderivative for
f(x)=x5+2x4x\displaystyle f(x)=\frac{x^5+2\sqrt{x}}{4x}
Find an antiderivative for
f(x)=x5+2x4x\displaystyle f(x)=\frac{x^5+2\sqrt{x}}{4x}
Find ∫ln(e4x)dx\displaystyle \int ln(e^{4x})dx
If F(x)=ln⁡(2x2)F(x)=\ln(2x^2) is an antiderivative of f(x)f(x), find f.
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.
Practice: Indefinite Simplify First
Q:\textbf{Q:} Find ∫[x2−4xx+x(2x2−3x)]dx\displaystyle\int_{ }^{ }\left[\frac{x^2-4\sqrt{x}}{\sqrt{x}}+x\left(2x^2-3x\right)\right]dx
Find ∫(x23−x5+1x34)dx\displaystyle \int \left(\sqrt[3]{x^2}-\sqrt[5]{x}+\frac{1}{\sqrt[4]{x^3}}\right)dx
Antiderivatives: Indefinite integrals
Evaluate the indefinite integral ∫x23−x5+1x34dx\displaystyle \int \sqrt[3]{x^2}-\sqrt[5]{x}+\frac{1}{\sqrt[4]{x^3}}dx.
Indefinite Integrals: Antiderivatives
Find the indefinite integral ∫2x(2x+3)2dx\int2x(2x+3)^2dx
Find ∫xe3 dx\int_{ }^{ }xe^3\ dx
Antiderivatives: Indefinite Integrals
Find ∫[x2−4xx+x(2x2−3x)]dx\int_{ }^{ }\left[\frac{x^2-4\sqrt{x}}{\sqrt{x}}+x\left(2x^2-3x\right)\right]dx.
Practice: Definite Integral
Practice: Definite Integral
Evaluate ∫−20∣x2−1∣dx\displaystyle \int_{-2}^0\left|x^2-1\right|dx.
Antiderivatives: Indefinite Integrals
For the following function, find the most general antiderivative.
f(x)=x5+2x4x\displaystyle f(x)=\frac{x^5+2\sqrt{x}}{4x}
Find the antiderivative FF
f(x)=4x+x22+5,F(1)=1/3\displaystyle f(x)=4x+\frac{x^2}{2}+5, F(1)=1/3
More Antiderivatives of 1x\frac{1}{x} Questions:
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
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