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# 6 Indefinite Integrals

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Indefinite Integrals
Bridge Course, August 2012

Definition

Let f(x) be a function. If F(x) is another function such that its derivative is equal to f(x), that is, then F(x) is called the primitive function. Note that primitive function is not unique. For example, All are primitive function of 2x.

In general, if F(x) is a primitive function of f(x) and C is a constant, then If F(x) is one of the primitive function of f(x), then we have This process of solving for F(x) from a giving f(x) is called indefinite integration and its opposite function is called differentiation, which is the process of finding a derivative.

Properties Examples                                                                                                           MAXIMA code

1. Evaluate integrate(2/x, x) 2. Evaluate integrate((x + 1) ^ 2/x, x) Method of Substitution

Let u = (x) be a differentiable function and . Then, Note that we can simply write as: Examples

1. Evaluate integrate(cos(3 ∗ x), x)

Let u = 3x and du = 3dx

Hence, 2. Evaluate integrate(x ^ 2 ∗ (2 ∗ x ^ 3 + 5) ^ 5, x)

Let u = 2 − x and du = −dx

Hence, 3. Evaluate integrate(x ^ 2 ∗ (2 ∗ x ^ 3 + 5) ^ 5, x)

Let u = 2 + 3 and du = 6 dx

Hence, Exercises

Evaluate the following integrals by method of substitution

1. integrate(x ^ 2 ∗ sqrt(x − 3), x)

2. integrate((x + 1) ∗ sin(x ^ 2 + 2 ∗ x + 1), x)

3. integrate(%e ^ x ∗ (1 − %e ^ x) ^ (−1/4), x)

4. integrate(2/sqrt(x) ∗ %e ^ (−sqrt(x)), x)

5. integrate(1/sqrt(a ^ 2 − x ^ 2), x)

6. integrate(1/(a ^ 2 + x ^ 2), x)

Integration of Rational Functions

A rational function is a quotient of two polynomials P(x) and Q(x).When integrating the rational function, the usual technique we use is to decompose the rational function into partial fractions.

Note that: We have to ensure f(x) is a proper fraction(that is degree of P(x) must be less that Q(x)) before we decompose f(x) into partial fractions.

Some transformation of partial fractions Examples

1. Evaluate integrate(1/(x ^ 2 − a ^ 2), x)

First, we have to decompose it into partial fractions, and we let By solving, we get .                                       partfrac(1/(x ^ 2 − a ^ 2), x)

Then, 2. Evaluate integrate((4 ∗ x ^ 2 + x + 12)/(x ∗ (x ^ 2 + 4)), x)

First, decompose into partial fractions, we let By solving, we get partfrac((4 ∗ x ^ 2+x+12)/(x ∗ (x ^ 2+4)), x)

Then, 3. Evaluate integrate((x ^ 2)/(x ^ 2 − 2 ∗ x + 1), x)

Since , we let By solving, we get partfrac((x ^ 2)/(x ^ 2−2 ∗ x+1), x)

Then, Exercises

Evaluate the following integrals

1. integrate(x/(2 ∗ x ^ 2 + x − 3), x)

2. find it yourself

3. find it yourself

4. find it yourself

5. find it yourself

Integration by parts

Let u(x) and v(x) be two real-valued functions with continuous first derivatives. Then The formula is simple so let’s get some illustration.

Examples

1. Evaluate integrate(x ^ 2 ∗ %e ^ x, x) 2. Evaluate integrate(x ∗ sin(2 ∗ x), x) 3. Evaluate integrate(x ^ n ∗ log(x), x) 4. Evaluate integrate(cos(x) ∗ %e ^ x, x) Hence, we have where C and C' are constants.

Exercises

Evaluate the following integrals

1. integrate(x ∗ %e ^ (2 ∗ x), x)

2. find it yourself

3. find it yourself

4. find it yourself

5. find it yourself