Integrals giving logarithms; exponentials and tangent

Warm-up

Three quick questions from earlier lessons.

Discover

In $\displaystyle\int \dfrac{2x}{x^2 + 1}\,dx$ , the top is the derivative of the bottom. With $u = x^2 + 1$ it becomes $\displaystyle\int \dfrac{du}{u}$ . What is that?

$\ln|u| + C$
$\dfrac{u^0}{0} + C$

Key idea

Three patterns come up again and again.

\displaystyle\int \dfrac{g'(x)}{g(x)}\,dx = \ln|g(x)| + C

;

\displaystyle\int e^{kx}\,dx = \dfrac{e^{kx}}{k} + C

;

\displaystyle\int \tan x\,dx = -\ln|\cos x| + C

Worked example

$\displaystyle\int \dfrac{\cos x}{\sin x + 3}\,dx$ .

u = \sin x + 3

du = \cos x\,dx

The top is the derivative of the bottom.

\ln(\sin x + 3) + C

ln|u|; sin x + 3 > 0.

Your turn, guided

Your turn: $\displaystyle\int e^{5x}\,dx$ .

\dfrac{e^{5x}}{5} + C

Divide by 5.

Practice ★

Find $F$ with $F'(x) = e^{ 2x}$ and $F(0) = 0$ .
Find $F$ with $F'(x) = e^{ 5x}$ and $F(0) = 0$ .

Evaluate $\displaystyle\int_0^1 \dfrac{2x}{x^2 + 2}\,dx$ .
Evaluate $\displaystyle\int_0^1 \dfrac{2x}{x^2 + 9}\,dx$ .

Watch out!

Practice ★★

Find $F$ with $F'(x) = \tan x$ on $\left(-\tfrac{\pi}{2}, \tfrac{\pi}{2}\right)$ and $F(0) = 3$ .
Find $F$ with $F'(x) = \tan x$ on $\left(-\tfrac{\pi}{2}, \tfrac{\pi}{2}\right)$ and $F(0) = -4$ .

Exit ticket

Three questions to check you have got it.

Find $F$ with $F'(x) = e^{ 2x}$ and $F(0) = 0$ .
Find $F$ with $F'(x) = e^{ 5x}$ and $F(0) = 0$ .
Find $F$ with $F'(x) = e^{ 4x}$ and $F(0) = 0$ .

Summary card

Useful patterns

$\dfrac{g'}{g} \to \ln|g|$ .
$e^{kx} \to \dfrac{e^{kx}}{k}$ .
$\tan x \to -\ln|\cos x|$ .

Watch out: a minus sign from $du$ .

End of the lesson

Next lesson: Algebra before integrating: division and completing the square Back to the chapter

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