12Week 47

12.1 Readings for this week's lectures

Sections 10.7, 12.1 – 12.3, 12.5 in the textbook.

12.2 Readings for this week's exercises

Sections 10.5-10.6 in the textbook.

12.3 Notes

Integration of a sum or difference of two functions

$\int f(x) \pm g(x) \, {\mathrm{d}}x = \int f(x) \pm \int g(x) \, {\mathrm{d}}x$

Integration by parts (indefinite integral)

$\int f(x)g'(x) \, {\mathrm{d}}x = f(x)g(x) - \int f'(x)g(x) \, {\mathrm{d}}x$

Integration by parts (definite integral)

$\int_a^b f(x)g'(x) \, {\mathrm{d}}x = \bigg|_a^b f(x)g(x) - \int_a^b f'(x)g(x) \, {\mathrm{d}}x$

Integration by substitution (indefinite integral)

Let $u=g(x)$ , then:

$\int f(g(x)) g'(x) \, {\mathrm{d}}x = \int f(u) \, {\mathrm{d}}u$

Integration by substitution (definite integral)

Let $u=g(x)$ , then:

$\int_a^b f(g(x)) g'(x) \, {\mathrm{d}}x = \int_{g(a)}^{g(b)} f(u) \, {\mathrm{d}}u$

Integration by substitution (general method)

Solve $\displaystyle\int G(x)\, dx$ by...

choosing a part of the integrand $u=g(x)$
find $du = g'(x)\, dx$
rewrite $\displaystyle\int G(x)\, dx = \int f(u)\, du$
find the indefinite integral $F(u)$
insert $g(x)$ again: $\displaystyle\int G(x)\, dx = F(g(x)) + C$

12.4 Problems

Below are some potential steps to apply in integration by parts for an indefinite integral. Arrange them in order by dragging and dropping from the left to the right to give a correct recipe for integration by parts for the integral

$\int f(x) g'(x) \, \mathrm{d} x$

Check that the new integrand is simpler than the original.

Compute $\mathrm{d} u = g'(x) \mathrm{d} x$ .

Identify the inner function in a composite function as $u$ .

Identify $f(x)$ as a function whose indefinite integral is known.

Identify $g'(x)$ as a function whose indefinite integral is known.

The first term is $f(x) g(x)$ , where $g'(x)$ has been integrated.

The second term is $-\displaystyle\int f'(x) g(x) \, \mathrm{d} x$ .

Identify $f(x)$ as a function whose derivative can be computed.

Solve the integral $-\displaystyle\int f'(x) g(x) \, \mathrm{d} x$ .

Use integration by parts to evaluate the following indefinite integrals:

$\displaystyle\int x \ln(x)\, {\mathrm{d}}x$
$\displaystyle\int 4x e^{4x}\, {\mathrm{d}}x$
$\displaystyle\int \frac{4x+2}{e^x}\, {\mathrm{d}}x$

Use integration by parts to evaluate the following definite integrals:

$\displaystyle\int_0^1 x^2e^x\, {\mathrm{d}}x$
$\displaystyle\int_0^2 (x-2)e^{x/2}\, {\mathrm{d}}x$
$\displaystyle\int_{-1}^1 x \ln(x+2)\, {\mathrm{d}}x$

Here you also need to use substitution.

Last week (exercise 11.14) you found the following definite integral:

$\int_{-3}^{3}(x-1)^2\, {\mathrm{d}}x$ This time find the integral by using integration by substitution.

Use integration by substitution to evaluate the following indefinite integral:

$\int 2x(x^2+1)^2\, {\mathrm{d}}x$

Use integration by substitution to evaluate the following definite integral:

$\int_5^8\frac{x}{x-4}\, {\mathrm{d}}x$

(Previous exam problem)

An integral is given by:

$\int_{0}^{1} x(1+x)^3\, {\mathrm{d}}x$

Calculate the integral.

(Previous exam problem)

Calculate the integral:

$\int x\sqrt{1+x^2}\, {\mathrm{d}}x$