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Integrals of $e^x$ and of $\frac{1}{x}$

Exponential and logarithmic functions, grade 12 (4 units) Lesson 17 of 20Learning

Integrals of $e^x$ and of $\frac{1}{x}$

Reminder: the antiderivative and the integral

From grade 11:

  • F\displaystyle F is an antiderivative of f\displaystyle f when F′(x)=f(x)\displaystyle {F'(x)}={f(x)}.
  • A function has infinitely many antiderivatives, a constant apart. All of them are written as the indefinite integral, with +C\displaystyle +C: ∫3x2 dx=x3+C\displaystyle {\int 3x^2\,dx}={x^3+C}.
  • A definite integral is a number: the difference between the values of the antiderivative at the limits.
∫abf(x) dx=F(b)−F(a)\displaystyle \int_{a}^{b}f(x)\,dx=F(b)-F(a)

An antiderivative is found by running a rule of differentiation backwards: each new derivative from the lessons before gives a new integral.

The integrals of ex\displaystyle e^x and of emx+b\displaystyle e^{mx+b}

(ex)′=ex\displaystyle {(e^x)'}={e^x}, so ex\displaystyle e^x is an antiderivative of itself. In emx+b\displaystyle e^{mx+b} the inner derivative is m\displaystyle m, so we divide by m\displaystyle m:

∫ex dx=ex+C\displaystyle \int e^{x}\,dx=e^{x}+C
∫emx+b dx=emx+bm+C\displaystyle \int e^{mx+b}\,dx=\frac{e^{mx+b}}{m}+C

The example below finds the integral of a sum, term by term:

∫(e2x+3)dx=e2x2+3x+C\int\left(e^{2x}+3\right)dx=\frac{e^{2x}}{2}+3x+C

A check by differentiating: (e2x2+3x)′=2e2x2+3=e2x+3\displaystyle {\left(\frac{e^{2x}}{2}+3x\right)'}={\frac{2e^{2x}}{2}+3}={e^{2x}+3}.

The integrals of 1x\displaystyle \frac{1}{x} and of 1mx+b\displaystyle \frac{1}{mx+b}

(ln⁡x)′=1x\displaystyle {(\ln x)'}={\frac{1}{x}} when x>0\displaystyle x>0, and also (ln⁡(−x))′=−1−x=1x\displaystyle {\left(\ln(-x)\right)'}={\frac{-1}{-x}}={\frac{1}{x}} when x<0\displaystyle x<0. Both cases together are written with an absolute value:

∫1x dx=ln⁡∣x∣+C\displaystyle \int\frac{1}{x}\,dx=\ln|x|+C
∫1mx+b dx=ln⁡∣mx+b∣m+C\displaystyle \int\frac{1}{mx+b}\,dx=\frac{\ln|mx+b|}{m}+C
  • ∫32x−1 dx=32ln⁡∣2x−1∣+C\displaystyle {\int\frac{3}{2x-1}\,dx}={\frac{3}{2}\ln|2x-1|+C}.
  • Where x>0\displaystyle x>0, ln⁡x\displaystyle \ln x may be written without the absolute value.
An antiderivative through a point

Of all the antiderivatives, only one passes through a given point. The example below finds the function F\displaystyle F whose derivative is F′(x)=e2x\displaystyle {F'(x)}={e^{2x}} and whose graph passes through (0,3)\displaystyle (0,3):

  • F(x)=e2x2+C\displaystyle {F(x)}={\frac{e^{2x}}{2}+C}.
  • We put in the point: F(0)=12+C=3\displaystyle {F(0)}={\frac{1}{2}+C}={3}, so C=52\displaystyle {C}={\frac{5}{2}}.
  • F(x)=e2x2+52\displaystyle {F(x)}={\frac{e^{2x}}{2}+\frac{5}{2}}.
The definite integral

The examples below calculate definite integrals with an antiderivative; the result is exact, with e\displaystyle e or ln⁡\displaystyle \ln:

∫01ex dx=[ex]01=e1−e0=e−1\int_{0}^{1}e^x\,dx=\left[e^x\right]_{0}^{1}=e^1-e^0=e-1∫1e1x dx=[ln⁡x]1e=ln⁡e−ln⁡1=1−0=1\int_{1}^{e}\frac{1}{x}\,dx=\left[\ln x\right]_{1}^{e}=\ln e-\ln1=1-0=1∫0ln⁡2ex dx=eln⁡2−e0=2−1=1\int_{0}^{\ln2}e^x\,dx=e^{\ln2}-e^0=2-1=1
The integral of a derivative

f\displaystyle f is an antiderivative of f′\displaystyle f', so the integral of a derivative is calculated without differentiating:

∫abf′(x) dx=f(b)−f(a)\displaystyle \int_{a}^{b}f'(x)\,dx=f(b)-f(a)

In the example below f(x)=(ln⁡x)2−4\displaystyle {f(x)}={(\ln x)^2-4}:

∫1ef′(x) dx=f(e)−f(1)=(1−4)−(0−4)=1\int_{1}^{e}f'(x)\,dx=f(e)-f(1)=(1-4)-(0-4)=1
Exercise 1

Find the integrals.

a

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

    b

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

      c

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

        d

          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

          Exercise 2

          Find the integrals. In part b, first divide each term of the numerator by ex\displaystyle e^x.

          a

            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

            b

              / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

              / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

              Exercise 3

              The derivative of a function F\displaystyle F is F′(x)=ex−1x\displaystyle {F'(x)}={e^x-\frac{1}{x}}, for x>0\displaystyle x>0.
              The graph of F\displaystyle F passes through the point (1,e)\displaystyle (1,e).

              a

              Find all the antiderivatives of F′\displaystyle F'.

                / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                b

                Find C\displaystyle C, and write F(x)\displaystyle F(x).

                Exercise 4

                Calculate the integrals. Write an exact result.

                a

                  / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                  / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                  b

                    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                    c

                      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                      d

                        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                        Exercise 5

                        The function f(x)=xln⁡x\displaystyle {f(x)}={x\ln x} is given.
                        Calculate the integral without differentiating f\displaystyle f.

                        f(x)=xln⁡x\displaystyle f(x)=x\ln x

                          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                          Exercise 6

                          a\displaystyle a is a positive parameter, and ∫0aex dx=2\displaystyle {\int_{0}^{a}e^x\,dx}={2}.
                          Calculate a\displaystyle a: write an equation and solve it.