5.3 Trigonometric and Exponential Integrals - Introduction to Calculus

Trigonometric and Exponential Integrals

Trigonometric Functions

THEOREM

Trigonometric Functions

\displaystyle \begin{aligned}\int \sin(x) \ dx & = -\cos(x) + C \\ \int \cos(x) \ dx & = \sin(x) + C \\ \int \csc^2(x) \ dx & = -\cot(x) + C \\ \int \sec^2(x) \ dx & = \tan(x) + C \\ \int \sec(x)\tan(x) \ dx & = \sec(x) + C \\ \int \csc(x)\cot(x) \ dx & = -\csc(x) + C\end{aligned}

Exponential / Logarithmic

THEOREM

Exponential / Logarithmic

For exponential functions, a>0; the logarithmic formulas also require a≠1 and an integration interval excluding x=0.

\displaystyle \begin{aligned}\int a^x\ln(a) \ dx & = a^x + C \\ \int e^x\ln(e) \ dx & = e^x + C \\ \int {1 \over x\ln(a)} \ dx & = \log_{\scriptscriptstyle a}|x| + C \\ \int {1 \over x} \ dx & = \ln|x| + C\end{aligned}

Find the antiderivatives.

(a)

\displaystyle \int 4^x\ln(4) + 5e^x - {6 \over x} \ dx

(b)

\displaystyle \int 3^z \ dz

Sometimes we need to manipulate the integral a little bit before we can apply the rules.

Find the antiderivatives.

(a)

\displaystyle \int {2s^3 - 5s^4 \over 3s^2} \ ds

(b)

\displaystyle \int \left({1 \over x} + {1 \over x^2}\right)\left(3 + 2x^2\right) \ dx

QUESTION

On an interval excluding zero, which expression equals

\displaystyle \int 1/x\,dx

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Trigonometric and Exponential Integrals

Trigonometric Functions

THEOREM

Trigonometric Functions

\displaystyle \begin{aligned}\int \sin(x) \ dx & = -\cos(x) + C \\ \int \cos(x) \ dx & = \sin(x) + C \\ \int \csc^2(x) \ dx & = -\cot(x) + C \\ \int \sec^2(x) \ dx & = \tan(x) + C \\ \int \sec(x)\tan(x) \ dx & = \sec(x) + C \\ \int \csc(x)\cot(x) \ dx & = -\csc(x) + C\end{aligned}

Exponential / Logarithmic

THEOREM

Exponential / Logarithmic

For exponential functions, a>0; the logarithmic formulas also require a≠1 and an integration interval excluding x=0.

\displaystyle \begin{aligned}\int a^x\ln(a) \ dx & = a^x + C \\ \int e^x\ln(e) \ dx & = e^x + C \\ \int {1 \over x\ln(a)} \ dx & = \log_{\scriptscriptstyle a}|x| + C \\ \int {1 \over x} \ dx & = \ln|x| + C\end{aligned}

Find the antiderivatives.

(a)

\displaystyle \int 4^x\ln(4) + 5e^x - {6 \over x} \ dx

(b)

\displaystyle \int 3^z \ dz

Sometimes we need to manipulate the integral a little bit before we can apply the rules.

Find the antiderivatives.

(a)

\displaystyle \int {2s^3 - 5s^4 \over 3s^2} \ ds

(b)

\displaystyle \int \left({1 \over x} + {1 \over x^2}\right)\left(3 + 2x^2\right) \ dx

QUESTION

On an interval excluding zero, which expression equals

\displaystyle \int 1/x\,dx

Trigonometric and Exponential Integrals

Trigonometric and Exponential Integrals

Trigonometric Functions

Trigonometric Functions

Exponential / Logarithmic

Exponential / Logarithmic

Exponential and logarithmic integrals

(a)

(b)

Simplify before integrating

(a)

(b)

KNOWLEDGE QUIZ

Trigonometric and Exponential Integrals

Trigonometric and Exponential Integrals

Trigonometric Functions

Trigonometric Functions

Exponential / Logarithmic

Exponential / Logarithmic

Exponential and logarithmic integrals

(a)

(b)

Simplify before integrating

(a)

(b)

KNOWLEDGE QUIZ