Integration as the Reverse of Differentiation: Quick Revision

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

Antiderivatives

• Definition: $F$ is an antiderivative of $f$ when $F' = f$. The whole family is $\int f(x)\,dx = F(x) + C$.
• The constant: two antiderivatives differ by a constant. A point on the curve fixes $C$.
• Power rule: $\int x^n\,dx = \dfrac{x^{n+1}}{n+1} + C$ for $n \ne -1$, and $\int \dfrac{dx}{x} = \ln|x| + C$.
• Rules: constants and sums come out term by term. There is no rule for a product.

Substitution

• Method: put $u = g(x)$ and $du = g'(x)\,dx$. Replace every $x$, integrate, then return to $x$.
• Linear inside: $\int f(ax + b)\,dx = \dfrac{1}{a}F(ax + b) + C$.
• Top is the derivative of the bottom: $\int \dfrac{f'}{f}\,dx = \ln|f| + C$.
• Four to know: $\int \tan x\,dx = \ln|\sec x|$, $\int \cot x\,dx = \ln|\sin x|$, $\int \sec x\,dx = \ln|\sec x + \tan x|$, $\int \operatorname{cosec} x\,dx = \ln|\operatorname{cosec} x - \cot x|$.

Trigonometric identities

• Even powers: $\sin^2 x = \dfrac{1 - \cos 2x}{2}$ and $\cos^2 x = \dfrac{1 + \cos 2x}{2}$.
• Odd powers: keep one factor for $du$ and use $\sin^2 x + \cos^2 x = 1$.
• Products: $2\sin A\cos B = \sin(A + B) + \sin(A - B)$.
• Tangent: $\tan^2 x = \sec^2 x - 1$.

A worked example

Find $\displaystyle\int \dfrac{2x + 3}{x^2 + 3x + 1}\,dx$.

1. Look at the top. The derivative of the bottom $x^2 + 3x + 1$ is $2x + 3$, which is the top.
2. Use $f'/f$. The integral is $\ln|x^2 + 3x + 1| + C$.

Practice with new numbers

A program to try

The program checks that $F(x) = \dfrac{x^3}{3} - x^2 + 4$ has the slope $x^2 - 2x$ at four points.

Mistakes to avoid

• Forgetting $+ C$. The answer is a family, not one function.
• Forgetting the factor of the inside. $\int \cos 5x\,dx$ needs $\dfrac{1}{5}$.
• Mixing $x$ and $u$. Return to $x$ at the end.
• A wrong sign. $\int \sin x\,dx = -\cos x$.

Check yourself

1. Find $\displaystyle\int (6x^2 - 4)\,dx$.

2. Find $\displaystyle\int \dfrac{x}{x^2 + 4}\,dx$.

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

4. Write $\cos^2 x$ without a square.

5. A curve has the slope $3x^2$ and passes through $(1, 5)$. Find its equation.

Sources

• OpenStax, Calculus Volume 1, Chapter 5, sections 5.4 and 5.5.
• NCERT, Mathematics Part II for Class XII, Chapter 7, the sections on integration as the inverse process of differentiation and on substitution.