Integration of Exponential and Logarithmic Function
Definition of the Natural Logarithmic Function
The natural logarithmic function is defined by
$
\ln x=\int_1^x \frac{1}{t} d t, \quad x>0
$
If $u$ is a differentiable function such that $u \neq 0$,
$
\int \frac{1}{u} d u=\ln |\boldsymbol{u}|+C
$
Whether $u$ is positive or negative, the integral of $(1 / u) d u$ is $\ln |u|+C$.
Integration of Exponential Function
$
\int e^u d u=e^u+C
$
Example 1
- Evaluate $\int_1^e \frac{x^2+3}{x} d x$
▶️Answer/Explanation
Solution
$\begin{aligned} & \int_1^e \frac{x^2+3}{x} d x=\int_1^e\left(\frac{x^2}{x}+\frac{3}{x}\right) d x=\int_1^e x d x+\int_1^e \frac{3}{x} d x=\left[\frac{x^2}{2}\right]_1^e+[3 \ln x]_1^e \\ & =\left[\frac{e^2}{2}-\frac{1}{2}\right]+3[\ln e-\ln 1]=\frac{e^2}{2}-\frac{1}{2}+3=\frac{e^2}{2}+\frac{5}{2}\end{aligned}$
Example 2
- Evaluate $\int_0^{\pi / 4}\left(e^{\tan x}+2\right) \sec ^2 x d x$.
▶️Answer/Explanation
Solution
$
\int_0^{\pi / 4}\left(e^{\tan x}+2\right) \sec ^2 x d x
$
$=\int_0^1\left(e^u+2\right) d u$ Let $u=\tan x, d u=\sec ^2 x d x$.
When $x=0, u=\tan 0=0$.
When $x=\pi / 4, u=\tan \pi / 4=1$.
$\begin{aligned} & =\left[e^u+2 u\right]_0^1 \\ & =\left[\left(e^1+2\right)-\left(e^0+0\right)\right]=e+1\end{aligned}$
Example 3
- Evaluate $\int_e^{e^2} \frac{(\ln x)^2}{x} d x$.
▶️Answer/Explanation
Solution
$
\int_e^{e^2} \frac{(\ln x)^2}{x} d x
$
$=\int_1^2 u^2 d u$ Let $u=\ln x, d u=d x / x$.
When $x=e, u=\ln e=1$.
When $x=e^2, u=\ln e^2=2$.
$=\left[\frac{1}{3} u^3\right]_1^2=\frac{1}{3}\left[2^3-1^3\right]=\frac{7}{3}$
Example 4
- $
\int_1^3 \frac{x+3}{x^2+6 x} d x=
$
(A) $\ln \frac{3}{2}$
(B) $\frac{\ln 27-\ln 7}{2}$
(C) $\ln 3$
(D) $\frac{\ln 20-\ln 5}{2}$
▶️Answer/Explanation
Ans:B
Example 5
- $
\int_0^1 \frac{x}{e^{x^2}} d x=
$
(A) $e-1$
(B) $\left(1-\frac{1}{e}\right)$
(C) $\frac{1}{2}\left(1-\frac{1}{e}\right)$
(D) $\frac{1}{2}\left(1-\frac{1}{e^2}\right)$
▶️Answer/Explanation
Ans:C
Example 6
- $
\int_0^{\pi / 2} \cos x e^{\sin x} d x=
$
(A) $-e$
(B) $1-e$
(C) $\frac{e}{2}$
(D) $e-1$
▶️Answer/Explanation
Ans:D