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Calculus

Antiderivatives

Concept Quizzes

Antiderivative and Indefinite Integration
Integration of Algebraic Functions
Integration of Exponential Functions
Integration of Trigonometric Functions
Integration of Rational Functions
Integration of Logarithmic Functions
Integrals - Problem Solving

Challenge Quizzes

Antiderivatives: Level 2 Challenges

Integration of Logarithmic Functions

If $xf'(x)=24\ln x,$ what is the function $f(x)?$ (Use $C$ as the constant of integration.)

12(\ln x)^2+C

12\ln x+C

\frac{24\ln x}{x}+C

24(\ln x)^2+C

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If $f(x)$ is a function such that $f(x)=\int \frac{1}{x \sqrt{\ln x+3}} dx$ and $f(e)=21,$ what is the value of $f(e^{5})?$

-17+4\sqrt{2}

-17-4\sqrt{2}

17-4\sqrt{2}

17+4\sqrt{2}

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What is the indefinite integral $\int \frac{1}{x}\left(7(\ln x)^{6}+5\right) dx?$ (Use $C$ as the constant of integration.)

(\ln x)^{7}+5\ln x+C

7(\ln x)^{7}+5\ln x+C

6(\ln x)^{6}+5\ln x+C

(\ln x)^{6}+5\ln x+C

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Let $N = \displaystyle \int_{1}^{e^ 8} \ln \left( \dfrac {1}{x} \right) \, dx$ . If $N = -ae^b - c$ , where $a$ , $b$ and $c$ are positive integers and $e$ is the base of the natural logarithm, what is the value of $a + b + c$ ?

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If $f(x)$ is a function satisfying $f(x)=\int \frac{6(\ln x)^{5}}{x} dx$ and $f(e)=8,$ what is the value of $f(e^2)?$

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