Integrating standard functions, integration by substitution, integration by parts, integration using partial fractions, and definite integrals for areas.

What this dot point is asking

WJEC wants you to integrate the standard functions (powers, $\mathrm{e}^x$, $\dfrac{1}{x}$, $\sin x$, $\cos x$), and to use the three main techniques: integration by substitution, integration by parts, and integration using partial fractions. You also evaluate definite integrals for areas. These techniques are the toolkit for the differential equations in this unit and the applied work in Unit 4.

The answer

Standard integrals

Integration by substitution

Substitution reverses the chain rule. Choose a new variable $u$ (often the inside of a composite function), differentiate to get $\dfrac{du}{dx}$, and replace both the function and $dx$ so the integral is entirely in $u$.

Integration by parts

By parts reverses the product rule and is used for products like $x\mathrm{e}^x$, $x\sin x$ and $x\ln x$.

Choose $u$ to be the factor that gets simpler when differentiated (often the polynomial, or $\ln x$), and $\dfrac{dv}{dx}$ to be the factor you can integrate.

Examples in context

Example 1. Area under an exponential

The area under $y = \mathrm{e}^{2x}$ from $x = 0$ to $x = 1$ is $\displaystyle\int_0^1 \mathrm{e}^{2x}\,dx = \left[\tfrac{1}{2}\mathrm{e}^{2x}\right]_0^1 = \tfrac{1}{2}(\mathrm{e}^2 - 1) \approx 3.19$. The standard exponential integral, with the $\tfrac{1}{2}$ from the chain rule in reverse, gives the area directly.

Example 2. A logarithmic integral

$\displaystyle\int \dfrac{2x}{x^2 + 1}\,dx$ has the derivative of the denominator on top, so it equals $\ln(x^2 + 1) + c$. Recognising the $\dfrac{f'(x)}{f(x)}$ pattern shortcuts the substitution entirely.

Example 3. Choosing the technique

Faced with $\displaystyle\int x\sin x\,dx$, the integrand is a product of a polynomial and a trig function, so by parts is the route, with $u = x$. Faced with $\displaystyle\int \dfrac{1}{x^2 - 1}\,dx$, the denominator factorises as $(x-1)(x+1)$, so partial fractions is the route. Faced with $\displaystyle\int 2x(x^2+1)^5\,dx$, the inner function and its derivative are both present, so substitution is the route. Matching the integrand to the technique is half the battle.

Try this

Q1. Find $\displaystyle\int \mathrm{e}^{4x}\,dx$. [2 marks]

• Cue. $\dfrac{1}{4}\mathrm{e}^{4x} + c$.

Q2. Find $\displaystyle\int x\mathrm{e}^x\,dx$ by parts. [3 marks]

• Cue. $u = x$, $v = \mathrm{e}^x$: $x\mathrm{e}^x - \int \mathrm{e}^x\,dx = x\mathrm{e}^x - \mathrm{e}^x + c$.

Q3. Find $\displaystyle\int \dfrac{3x^2}{x^3 + 2}\,dx$. [2 marks]

• Cue. The top is the derivative of the bottom, so the integral is $\ln|x^3 + 2| + c$.