Calculus: Integrals III

Logarithmic Functions

Since the derivative of the natural log is $\dfrac{d}{dx}\ln(x) = \dfrac{1}{x}$, we can conclude the following:

$$\displaystyle\int \dfrac{1}{x} \, dx = \ln|x| + C$$

The integral of the natural log itself is as follows:

$$\displaystyle\int \ln(x) \, dx = x(\ln(x) - 1) + C$$

You will be able to derive the integral of the natural logarithm later in the section on integration by parts.

Problems

1. Evaluate: $\displaystyle\int \dfrac{1}{2x} \, dx$

$\displaystyle\int \dfrac{1}{2x} \, dx = \displaystyle\int \dfrac{1}{2}\dfrac{1}{x} \, dx \\ \displaystyle\int \dfrac{1}{2x} \, dx = \dfrac{1}{2}\displaystyle\int \dfrac{1}{x} \, dx \\ \displaystyle\int \dfrac{1}{2x} \, dx = \dfrac{1}{2}(\ln|x| + c) \\ \displaystyle\int \dfrac{1}{2x} \, dx = \dfrac{1}{2}\ln|x| + \dfrac{1}{2}c \\ \displaystyle\int \dfrac{1}{2x} \, dx = \dfrac{1}{2}\ln|x| + c \\$

Don't forget that half of an arbitrary constant is still just an arbitrary constant.

2. Evaluate: $\displaystyle\int \dfrac{x+4}{x} -1 \, dx$

$\displaystyle\int \dfrac{x+4}{x} - 1 \, dx = \displaystyle\int \dfrac{x}{x} + \dfrac{4}{x} - 1 \, dx \\ \displaystyle\int \dfrac{x+4}{x} - 1 \, dx = \displaystyle\int 1 + \dfrac{4}{x} - 1 \, dx \\ \displaystyle\int \dfrac{x+4}{x} - 1 \, dx = \displaystyle\int \dfrac{4}{x} \, dx \\ \displaystyle\int \dfrac{x+4}{x} - 1 \, dx = 4\displaystyle\int \dfrac{1}{x} \, dx \\ \displaystyle\int \dfrac{x+4}{x} - 1 \, dx = 4\left( \ln|x| + c \right) \\ \displaystyle\int \dfrac{x+4}{x} - 1 \, dx = 4\ln|x| + 4c \\ \displaystyle\int \dfrac{x+4}{x} - 1 \, dx = 4\ln|x| + c \\$

3. Evaluate: $\displaystyle\int \dfrac{9001}{x} \, dx$

$\displaystyle\int \dfrac{9001}{x} \, dx = 9001\displaystyle\int \dfrac{1}{x} \, dx \\ \displaystyle\int \dfrac{9001}{x} \, dx = 9001(\ln(x) + c) \\ \displaystyle\int \dfrac{9001}{x} \, dx = 9001\ln(x) + c \\$

4. Evaluate: $\displaystyle\int \log_2(x) \, dx$

$\displaystyle\int \log_2(x) \, dx = \displaystyle\int \dfrac{\ln(x)}{\ln(2)} \, dx \\ \displaystyle\int \log_2(x) \, dx = \dfrac{1}{\ln(2)}\displaystyle\int \ln(x) \, dx \\ \displaystyle\int \log_2(x) \, dx = \dfrac{1}{\ln(2)}(x(\ln(x)-1) + c) \\ \displaystyle\int \log_2(x) \, dx = \dfrac{x(\ln(x)-1)}{\ln(2)} + c \\$

5. Evaluate: $\displaystyle\int \log_{10}(x) \,dx$

$\displaystyle\int \log_{10}(x) \,dx = \displaystyle\int \dfrac{\ln(x)}{\ln(10)} \, dx \\ \displaystyle\int \log_{10}(x) \,dx = \dfrac{1}{\ln(10)}\displaystyle\int \ln(x) \, dx \\ \displaystyle\int \log_{10}(x) \,dx = \dfrac{1}{\ln(10)}(x(\ln(x)-1) + c) \\ \displaystyle\int \log_{10}(x) \,dx = \dfrac{x(\ln(x)-1)}{\ln(10)} + c \\$

6. Evaluate: $\displaystyle\int \ln(ex) - 1\, dx$
$\displaystyle\int \ln(ex) - 1 \, dx = \displaystyle\int \ln(x) + \ln(e) - 1 \, dx \\ \displaystyle\int \ln(ex) - 1 \, dx = \displaystyle\int \ln(x) + 1 - 1 \, dx \\ \displaystyle\int \ln(ex) - 1 \, dx = \displaystyle\int \ln(x) \, dx \\ \displaystyle\int \ln(ex) - 1 \, dx = x(\ln(x)-1) + c \\$
7. Evaluate: $\displaystyle\int \ln\left(x^3\right) \, dx$

$\displaystyle\int \ln\left(x^3\right) \, dx = \displaystyle\int 3\ln(x) \, dx \\ \displaystyle\int \ln\left(x^3\right) \, dx = 3\displaystyle\int \ln(x) \, dx \\ \displaystyle\int \ln\left(x^3\right) \, dx = 3(x(\ln(x)-1) + c) \\ \displaystyle\int \ln\left(x^3\right) \, dx = 3x(\ln(x)-1) + c \\$

8. Evaluate: $\displaystyle\int \dfrac{e}{e^{\ln(x)}} \, dx$

$\displaystyle\int \dfrac{e}{e^{\ln(x)}} \, dx = \displaystyle\int \dfrac{e}{x} \, dx \\ \displaystyle\int \dfrac{e}{e^{\ln(x)}} \, dx = e\displaystyle\int \dfrac{1}{x} \, dx \\ \displaystyle\int \dfrac{e}{e^{\ln(x)}} \, dx = e(\ln|x| + c) \\ \displaystyle\int \dfrac{e}{e^{\ln(x)}} \, dx = e\ln|x| + c \\$

9. Evaluate: $\displaystyle\int \ln(\ln(e^x)) \, dx$

$\displaystyle\int \ln(\ln(e^x)) \, dx = \displaystyle\int \ln(x) \, dx \\ \displaystyle\int \ln(\ln(e^x)) \, dx = x(\ln(x) -1 ) + c \\$

10. Evaluate: $\displaystyle\int \ln\left(\cos^2(x)\right) + \ln\left(x + x\tan^2(x)\right) \, dx$

$\displaystyle\int \ln\left(\cos^2(x)\right) + \ln\left(x + x\tan^2(x)\right) \, dx = \displaystyle\int \ln(\left( \cos^2(x)\left(x + x\tan^2(x)\right)\right) \, dx \\ \displaystyle\int \ln\left(\cos^2(x)\right) + \ln\left(x + x\tan^2(x)\right) \, dx = \displaystyle\int \ln(\left( x\cos^2(x) + x\cos^2(x)\tan^2(x)\right) \, dx \\ \displaystyle\int \ln\left(\cos^2(x)\right) + \ln\left(x + x\tan^2(x)\right) \, dx = \displaystyle\int \ln(\left( x\cos^2(x) + x\sin^2(x)\right) \, dx \\ \displaystyle\int \ln\left(\cos^2(x)\right) + \ln\left(x + x\tan^2(x)\right) \, dx = \displaystyle\int \ln(\left( x\left(\cos^2(x) + \sin^2(x)\right)\right) \, dx \\ \displaystyle\int \ln\left(\cos^2(x)\right) + \ln\left(x + x\tan^2(x)\right) \, dx = \displaystyle\int \ln(x(1)) \, dx \\ \displaystyle\int \ln\left(\cos^2(x)\right) + \ln\left(x + x\tan^2(x)\right) \, dx = \displaystyle\int \ln(x) \, dx \\ \displaystyle\int \ln\left(\cos^2(x)\right) + \ln\left(x + x\tan^2(x)\right) \, dx = x(\ln(x) - 1) + c \\$