Calculate the following definite integrals:

1. int_(-2)^(+2) sqrt(4 − x^2) dx
2. int_0^pi sin(x) dx
3. int_0^1 xe^x dx
Approach

Take the integral: int sqrt(4-x^2) dx For the integrand sqrt(4-x^2), substitute x = 2 sin(u) and dx = 2 cos(u) du. Then sqrt(4-x^2) = sqrt(4-4 sin^2(u)) = 2 cos(u) and u = sin^(-1)(x/2): = 4 int cos^2(u) du

Write cos^2(u) as 1/2 cos(2 u)+1/2: = 4 int (1/2 cos(2 u)+1/2) du

Integrate the sum term by term and factor out constants: = 2 int 1 du+2 int cos(2 u) du

For the integrand cos(2 u), substitute s = 2 u and ds = 2 du: = int cos(s) ds+2 int 1 du

The integral of cos(s) is sin(s): = sin(s)+2 int 1 du

The integral of 1 is u: = sin(s) + 2 u + "constant"

Substitute back for s = 2 u: = 2 u + sin(2 u) + "constant"

Apply the double angle formula sin(2 u) = 2 cos(u) sin(u): = 2 u + 2 sin(u) cos(u) + "constant"

Express cos(u) in terms of sin(u) using cos^2(u) = 1-sin^2(u): = 2 u + 2 sin(u) sqrt(1-sin^2(u)) + "constant"

Substitute back for u = sin^(-1)(x/2): = 1/2 sqrt(4-x^2) x + 2 sin^(-1)(x/2) + "constant"

int_(-2)^(+2) sqrt(4 − x^2) dx = (1/2 * 0 + 2 sin^(-1)(1) + "constant") - (1/2 * 0 + 2 sin^(-1)(-1) + "constant") = pi + pi = 2pi

int sin(x) dx = -cos(x) + "constant"

int_0^pi sin(x) dx = (-cos(pi) + "constant") - (-cos(0) + "constant") = 1 - (-1) = 2

int e^x x dx

For the integrand e^x x, integrate by parts, int f dg = f g - int g d f, where f = x, dg = e^x dx, df = dx, g = e^x: = e^x x - int e^x dx

The integral of e^x is e^x: = e^x x - e^x + "constant"

Which is equal to: = e^x (x-1) + "constant"

int_0^1 xe^x dx = e^1(1 - 1) - e^0(0 - 1) = 1

Solution
1. 2pi
2. 2
3. 1
• URL:
• Language:
• Subjects: math
• Type: Calculate
• Duration: 30min
• Credits: 6
• Difficulty: 0.6
• Tags: hpi integral definite integral
• Note:
HPI, 2014-05-26, Mathe 2, Aufgabe 30