MTH6131 Chap.13 Integration Formulas and Substitution
Integration Formulas and Substitution
Integration reverses differentiation
The schedule opens the second half of the course with an introductory lecture on integration, followed by three weeks of techniques. An antiderivative of f is any function whose derivative is f, and because any two antiderivatives differ by a constant, the indefinite integral is written with plus C.
The core table follows directly from the derivatives of the earlier chapters: powers, the reciprocal, exponentials, the six trigonometric forms, and the inverse trigonometric forms.
The one exception in the power rule
The integral of u to the power n is u to the power n + 1 over n + 1, except when n = −1, where it would divide by zero. That case is the integral of 1/u, which is ln|u|.
The absolute value matters because 1/u is defined for negative u as well.
Adjusting the differential
The exercise solutions for this topic rarely name a new variable. Instead they rewrite dx as a multiple of the differential of the inner expression, so that the integral matches a line of the table exactly. For the square root of 2x − 5, for example, dx is half of d(2x − 5), and the power rule finishes the job.
The rule is strict: only a constant factor may be supplied from outside the integral. If the differential is short by a power of x, a different substitution or a different method is needed.
Rewrites before substitution
Two preliminary moves appear throughout the exercises.
A product that can be expanded is multiplied out and integrated term by term, and a quotient whose numerator has degree at least that of the denominator is divided first. After that, recognising the shape of the integrand, a numerator that is the derivative of the denominator, an exponential of a function whose derivative is present, a constant over a sum or difference of squares, decides which line of the table applies.
What this chapter covers
- 01
Antiderivatives and the constant of integration
- 02
The core table of standard integrals
- 03
The power rule and its n = −1 exception
- 04
Adjusting the differential by a constant
- 05
Expanding and dividing before integrating
- 06
Log and exponential forms
- 07
Trigonometric powers by substitution
- 08
Arctangent, arcsine and log-quotient forms
Worked example · free
A numerator that is half the derivative
- 2Compare the numerator with the derivative of the denominator.
- 2Adjust by a constant and apply the logarithm form.
Key terms
- Indefinite Integral
- The family of all antiderivatives of a function, written with a constant of integration.
- Constant of Integration
- The arbitrary constant C added to an antiderivative, because functions differing by a constant share a derivative.
- Substitution
- Rewriting an integral in terms of an inner expression and its differential so that it matches a standard form.
- Standard Form
- One line of the table of known integrals, such as the integral of du over u equals ln of the absolute value of u.
Integration Formulas and Substitution FAQ
Why do I have to add plus C to an indefinite integral?
Because the derivative of a constant is zero, so if F is an antiderivative of f, so is F plus any constant. The indefinite integral names that whole family. Leaving out plus C gives only one member and loses the information needed later, for example to fit an initial condition in a differential equation.
How do I know which substitution to make?
Look for an inner expression whose derivative, up to a constant factor, appears elsewhere in the integrand. If it is there, adjust the differential by that constant and use the table. If the derivative is missing a power of the variable, that substitution will not work and another approach is needed.
How can I tell an arctangent integral from a logarithm integral?
Look at the denominator after completing the square. A constant over a sum of squares, such as 4 plus x squared, gives an arctangent. A constant over a difference of squares, such as x squared minus 16, gives a logarithm of a quotient. A numerator that is the derivative of the denominator gives a plain logarithm whatever the denominator is.
Exam move
Learn the core table by writing it out from the derivative list rather than memorising it separately. Then work integrals in short timed sets, and differentiate every answer; the check is the fastest way to discover whether an adjustment of the differential was right.
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