Methods of Integration: Quick Revision

A brief note on Methods of Integration. Read it before a test. Try each question at the end before you open its answer.

Special forms

• Inverse tangent: $\int \dfrac{dx}{x^2 + a^2} = \dfrac{1}{a}\tan^{-1}\dfrac{x}{a} + C$.
• Inverse sine: $\int \dfrac{dx}{\sqrt{a^2 - x^2}} = \sin^{-1}\dfrac{x}{a} + C$.
• Logarithms: $\int \dfrac{dx}{x^2 - a^2} = \dfrac{1}{2a}\ln\left|\dfrac{x - a}{x + a}\right| + C$ and $\int \dfrac{dx}{\sqrt{x^2 \pm a^2}} = \ln\left|x + \sqrt{x^2 \pm a^2}\right| + C$.
• A quadratic: complete the square, then put $u = x + \dfrac{b}{2a}$.
• A linear top: write $px + q = A(2ax + b) + B$. The first part gives a logarithm.

Partial fractions

• Forms: $\dfrac{A}{x - a}$ for a linear factor, all powers for a repeated factor, and $\dfrac{Bx + C}{x^2 + p^2}$ for a quadratic factor.
• Cover-up rule: hide a linear factor and put its root in the rest, to find its constant.
• Improper fractions: if the top has degree at least that of the bottom, divide first.
• Integrals: $A\ln|x - a|$ and $-\dfrac{B}{x - a}$.

Integration by parts

• Formula: $\int u\,dv = uv - \int v\,du$.
• Choice of $u$: ILATE, which is Inverse trigonometric, Logarithmic, Algebraic, Trigonometric, Exponential.
• Alone: $\ln x$ and $\tan^{-1}x$ use $dv = dx$.
• A cycle: for $e^x\sin x$, use parts twice and solve for the integral.
• A shortcut: $\int e^x\bigl[f(x) + f'(x)\bigr]dx = e^x f(x) + C$.

A worked example

Find $\displaystyle\int x\,e^{x}\,dx$.

1. Choose. Take $u = x$ and $dv = e^x\,dx$, so $du = dx$ and $v = e^x$.
2. Apply the formula. $x\,e^x - \int e^x\,dx = x\,e^x - e^x + C$.

Practice with new numbers

A program to try

The program checks the split of $\dfrac{1}{(x - 1)(x + 2)}$ into two fractions at four values of $x$.

Mistakes to avoid

• Wrong constant in front. $\dfrac{1}{a}$ for the inverse tangent, and $\dfrac{1}{2a}$ for the logarithm.
• Not dividing first. An improper fraction cannot be split directly.
• A missing term. A repeated factor needs every power.
• A lost sign in parts. The formula is $uv - \int v\,du$.

Check yourself

1. Find $\displaystyle\int \dfrac{dx}{x^2 + 36}$.

2. Split $\dfrac{5}{(x - 2)(x + 3)}$ into partial fractions.

3. Find $\displaystyle\int x\cos x\,dx$.

4. Find $\displaystyle\int e^x\left(\cos x - \sin x\right)dx$.

5. Which part comes first in ILATE, a logarithm or a power of $x$?

Sources

• OpenStax, Calculus Volume 2, Chapter 3, sections 3.1 and 3.4.
• NCERT, Mathematics Part II for Class XII, Chapter 7, the sections on integrals of some particular functions, partial fractions and parts.