Integrate the exponential, reciprocal and trigonometric functions and their linear composites as the reverse of the corresponding derivatives

What this dot point is asking

SEAB wants you to integrate the standard functions beyond powers: the exponential $e^x$, the reciprocal $\dfrac{1}{x}$, and the trigonometric functions $\sin x$ and $\cos x$, together with their linear composites such as $e^{2x}$ or $\cos 3x$. Each is the reverse of a derivative you already know.

The answer

The standard integrals

Reverse the standard derivatives:

Note the minus sign when integrating sine, and the modulus in the logarithm.

Why the reciprocal gives a logarithm

The power rule fails for $x^{-1}$ because it would divide by zero. Instead, since $\dfrac{d}{dx}\ln x = \dfrac{1}{x}$, the integral of $\dfrac{1}{x}$ is $\ln|x|$. The modulus allows for negative $x$.

Linear composites

For a function of $(ax + b)$, integrate as the simple version and divide by the coefficient $a$ of $x$ inside, just as with powers:

The division by $a$ undoes the chain-rule factor differentiation would have introduced.

Putting it together

Integrate a sum term by term, applying the right standard integral and the inner-coefficient division to each. Keep one constant of integration for the whole expression.

Definite integrals of these functions

The same standard results apply to definite integrals: integrate first, then substitute the upper and lower limits and subtract. For exponentials and trigonometric functions this often produces neat exact values, such as $\int_0^{\pi} \sin x\,dx = [-\cos x]_0^{\pi} = 2$, which examiners expect to be left exact rather than rounded.

Checking by differentiating

Since integration reverses differentiation, differentiate your answer to confirm it returns the integrand. This quickly exposes a dropped minus sign on a sine, a missing modulus on a logarithm, or a forgotten inner-coefficient division on a composite.

Examples in context

Example 1. Charge from a current. An electric current that varies as $I = I_0 e^{-t/\tau}$ integrates to give the total charge that has flowed, the exponential integral underpinning capacitor discharge.

Example 2. Displacement of an oscillator. A velocity varying as $v = \cos(\omega t)$ integrates to a sine displacement, so integrating trigonometric functions describes how an oscillating object moves over time.

Try this

Q1. Find $\displaystyle\int e^{3x}\,dx$. [2 marks]

• Cue. $\dfrac{1}{3}e^{3x} + C$.

Q2. Find $\displaystyle\int \cos 2x\,dx$. [2 marks]

• Cue. $\dfrac{1}{2}\sin 2x + C$.

Q3. Find $\displaystyle\int \dfrac{5}{x}\,dx$. [2 marks]

• Cue. $5\ln|x| + C$.