2.6.4 Practice set: integration techniques

Substitution

Integration by substitution (lesson 2.6.1)

0 of 6 correct

1.
Find $\int_0^2 2x\,e^{x^2}\,dx$
2.
Find $\int_0^1 (3x + 1)^2\,dx$
3.
Find an antiderivative of $x^2(x^3 + 1)^4$ (leave out $+ C$ )
4.
Find $\int_0^1 x\,e^{-x^2}\,dx$
5.
Find $\int_0^1 3x^2(x^3 + 1)^2\,dx$
6.
Which method suits $\int x\,e^x\,dx$ ? Type 1 for substitution, 2 for integration by parts

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

Integration by parts

Integration by parts (lesson 2.6.2)

0 of 6 correct

1.
Find $\int_0^2 x\,e^x\,dx$
2.
Find $\int_1^{e^3} \ln x\,dx$
3.
In $\int x\ln x\,dx$ , which factor should be $u$ ? Type 1 for $\ln x$ , 2 for $x$
4.
Find $\int_0^1 x\,e^{3x}\,dx$
5.
Find an antiderivative of $(\ln x)^2$ (leave out $+ C$ )
6.
Find $\int_0^1 x^2 e^{-x}\,dx$

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

Improper integrals

Improper integrals and the Gaussian integral (lesson 2.6.3)

0 of 6 correct

1.
Find $\int_1^{\infty} \frac{3}{x^2}\,dx$
2.
Find $\int_4^{\infty} \frac{1}{x^{3/2}}\,dx$
3.
Find $\int_0^{9} \frac{1}{\sqrt{x}}\,dx$ (improper at 0)
4.
Find $\int_0^{\infty} x\,e^{-2x}\,dx$
5.
Find $\int_{-\infty}^{\infty} e^{-(x - 3)^2}\,dx$ (exactly)
6.
Does $\int_{-1}^{2} \frac{1}{x^2}\,dx$ converge? Type 1 for yes, 0 for no

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

Coding exercise

Exercise

The normal distribution from its density

Write two functions without using scipy.stats.

normal_pdf(x) returns $\varphi(x) = \frac{1}{\sqrt{2\pi}}e^{-x^2/2}$ .
normal_cdf(x) returns $N(x) = \int_{-\infty}^{x}\varphi(t)\,dt$ , computed with scipy.integrate.quad and math.inf (lesson 2.6.3).

The tests compare your functions with scipy.stats.norm. They also check that $\varphi$ is symmetric, that $N(0) = \frac{1}{2}$ , that $N(x) + N(-x) = 1$ , and the far tails $N(-8)$ and $N(8)$ .

$N(-8)$ is about $6.2 \times 10^{-16}$ , so the test asks for it to be within $10^{-12}$ of 0. Case study 2.6.5 shows why such tiny tail areas need care.

See the tests

from scipy.stats import norm


def test_pdf_at_zero():
    """φ(0) = 1/√(2π), about 0.3989"""
    expect_close(normal_pdf(0.0), norm.pdf(0.0), what="normal_pdf(0)")


def test_pdf_symmetric():
    """φ is symmetric: φ(1.5) = φ(−1.5)"""
    expect_close(normal_pdf(-1.5), normal_pdf(1.5), what="normal_pdf(−1.5) vs normal_pdf(1.5)")


def test_cdf_middle():
    """N(0) = 0.5"""
    expect_close(normal_cdf(0.0), 0.5, what="normal_cdf(0)", rel=1e-8)


def test_cdf_values():
    """N(x) matches scipy.stats.norm.cdf at several points"""
    for x in (-2.0, -0.5, 1.0, 1.96):
        expect_close(normal_cdf(x), norm.cdf(x), what=f"normal_cdf({x})", rel=1e-7)


def test_cdf_symmetry():
    """N(x) + N(−x) = 1: the two tails of the bell have equal area"""
    for x in (0.4, 1.3, 2.5):
        expect_close(normal_cdf(x) + normal_cdf(-x), 1.0, what=f"normal_cdf({x}) + normal_cdf(−{x})", rel=1e-8)


def test_cdf_tails():
    """N(−8) is nearly 0 and N(8) nearly 1"""
    expect_close(normal_cdf(-8.0), 0.0, what="normal_cdf(−8)", abs_tol=1e-12, rel=0)
    expect_close(normal_cdf(8.0), 1.0, what="normal_cdf(8)", rel=1e-9)

Substitution

Integration by parts

Improper integrals

Coding exercise

Free further reading