MATH 12002 - CALCULUS I 5.3: Integrals and the Natural Exponential - - PowerPoint PPT Presentation

MATH 12002 - CALCULUS I §5.3: Integrals and the Natural Exponential Function

Professor Donald L. White

Department of Mathematical Sciences Kent State University

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Integral of ex

Recall the differentiation formula d dx ex = ex. Reversing this formula gives the integration formula

• ex dx = ex + C.

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Examples

Evaluate the integrals:

1

• x3ex4+5 dx.
• x3ex4+5 dx

=

• x3ex4+5 dx

Let u = x4 + 5 so du = 4x3 dx and 1

4du = x3 dx

= 1 4

• eu du

= 1 4eu + C = 1 4ex4+5 + C. Note: This integral can also be evaluated by letting u = ex4+5, so that du = 4x3ex4+5 dx and 1

4du = x3ex4+5 dx. The integral becomes

1

4 du = 1 4u + C = 1 4ex4+5 + C.

D.L. White (Kent State University) 3 / 7

Examples

2

• e−5x dx.

We can use a formal substitution, letting u = −5x, so that du = −5 dx and −1

5du = dx, and so

• e−5x dx = −1

5

• eu du = −1

5eu + C = −1 5e−5x + C. However, when u is just a constant multiple of x, it is usually more efficient to use the “guess and check” method: We would guess that

• e−5x is approximately e−5x.

Now check the derivative:

d dx e−5x = −5e−5x

and so we don’t get the function we’re integrating. So we multiply our “guess” by −1

5 to compensate, and check again: d dx (−1 5e−5x) = −1 5(−5e−5x) = e−5x.

Since

d dx (−1 5e−5x) = e−5x, we have that

• e−5x dx = −1

5e−5x + C.

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Examples

3

ln π

ln π

2

ex cos(ex) dx. ln π

ln π

2

ex cos(ex) dx = ln π

ln π

2

ex cos(ex) dx u = ex du = ex dx x = ln π ⇒ u = eln π = π x = ln π

2 ⇒ u = eln π

2 = π

2

= π

π 2

cos u du = sin u

• π

π/2

= sin π − sin π

2 = 0 − 1 = −1.

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Examples

4

• ex

(1 + ex)2 dx.

• ex

(1 + ex)2 dx =

• ex(1 + ex)−2 dx

Let u = 1 + ex, so du = ex dx =

• u−2 du = −u−1 + C = −

1 1 + ex + C.

5

• ex

1 + ex dx.

• ex

1 + ex dx =

• ex

1 + ex dx Let u = 1 + ex, so du = ex dx = 1 u du = ln |u| + C = ln(1 + ex) + C.

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Examples

6

1 + ex ex dx. 1 + ex ex dx =

• 1

ex + ex ex dx =

• e−x + 1 dx

= −e−x + x + C.

7

• e dx.

Since e is a constant,

• e dx = ex + C,

just like

• 3 dx = 3x + C or
• −735 dx = −735x + C.

Be sure to distinguish between the constant e ≈ 2.718281828459045 and the exponential function ex.

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