Examples from Seen questions

Average value of a piecewise function

The average value of a function $f$ on $[a,b]$ is

Compute the average $\bar f$ on $\left[-\frac{\pi}{4},\frac{\pi}{4}\right]$ for

Solution

The interval is split at $x=0$, so the integral of $f$ is the sum of one integral from each branch. Parts (a) and (b) find the two antiderivatives, part (c) applies them on the appropriate intervals and part (d) asks whether $f$ attains its average.

(a) The product-to-sum integral

The product-to-sum identity

applies after writing $\cos3x\sin2x$ as $\sin2x\cos3x$. Thus

The difference of arguments is $-x$ and $\sin(-x)=-\sin x$; this sign is easy to lose.

(b) The tangent-secant integral

Use the substitution

Then

For the definite integral on the second branch, the bounds transform as

Therefore

(c) Computing the average

For the first branch, use the antiderivative from (a):

The second branch contributes $\frac14$ by (b). Each formula for $f$ applies only on its own subinterval, so these two integrals give the total:

The interval length is

Consequently,

(d) Existence of a point attaining the average

The first branch is continuous on $\left[-\frac{\pi}{4},0\right]$. At the endpoints of this subinterval,

Also,

Thus $\bar f$ lies strictly between $f(0)$ and $f\left(-\frac{\pi}{4}\right)$. The Intermediate Value Theorem applied to the first branch gives a point

such that $f(c)=\bar f$. Therefore the answer is Yes.

Which branch owns the single point $x=0$ does not affect the integral or the average. What matters is which formula is integrated on each side.

Reduction formulae and substitution of a hyperbolic function

Consider

It can be written as a reduction formula in the form

where $a_n$ and $b_n$ depend on $n$. Find $a_n$ and $b_n$ in terms of $n$.

Next, consider

Suggest a substitution so that

and then write the rational integrand as

where the degree of each numerator is less than the degree of its denominator. Find $A(t)$, $B(t)$ and $C(t)$.

Solution

In the first integral, the factor $7+12x$ is the derivative of the exponent:

Differentiating a product with one lower power of $7+12x$ therefore produces both an $I_n$ term and an $I_{n-2}$ term.

(a) Finding the reduction formula

For $n\ge2$, differentiate

Using the product rule and the chain rule gives

Integrate from $0$ to $1$. The second term on the right is $I_n$, while the first term is $12(n-1)I_{n-2}$, so

At the endpoints,

and

Therefore

Comparing this with $I_n=a_n+b_nI_{n-2}$ gives

(b) Substitution and partial fractions

The numerator $\cosh(6x)$ is the derivative of $\sinh(6x)$ up to a factor of $6$, so this is an ordinary substitution, not a surd-clearing hyperbolic substitution. Set

Thus $\cosh(6x)\,dx=\frac16\,dt$ and

Factor the denominator:

The quadratic has discriminant

so it is an irreducible real quadratic. The required partial-fraction form is

The question writes the last numerator as $C(t)$; here $C(t)=Ct+D$.

Multiplying through by the denominator gives the polynomial identity

Since

substitution of $t=0$ gives

Substitution of $t=7$ gives

To find the quadratic numerator, expand the identity and compare the coefficients of $t$ and $t^2$:

The coefficient of $t$ gives

The coefficient of $t^2$ then gives

Hence the requested numerators are

As a check, the decomposition is

Consequently, on any interval that does not contain a point where $\sinh(6x)=0$ or $\sinh(6x)=7$, an antiderivative is

The first part is a reduction formula: differentiating a product with one lower power gives a recurrence that drops the index by two. The second is a substitution followed by partial fractions for a proper rational function.