A tutorial, with examples and detailed solutions, in using the rules of indefinite integrals in calculus is presented. A set of questions with solutions is also included.
In what follows, C is a constant of integration and can take any value.
Example: Evaluate the integral
\[ \int x^5 dx \]
Solution:
\[ \int x^5 dx = \dfrac{x^{5 + 1}}{ 5 + 1} + c = \dfrac{x^6}{6} + c \]
Example: Evaluate the integral
\[ \int 5 \sin \; x dx \]
Solution:
According to the above rule
\( \displaystyle \int 5 \sin (x) dx = 5 \int \sin(x) dx \)
\( \displaystyle \int \sin(x) dx \) is given by 2.1 in table of integral formulas, hence
Hence
\( \displaystyle \int 5 \sin(x) dx = - 5 \cos x + C \)
Example: Evaluate the integral
\[ \int (x + e^x) dx \]
Solution:
According to the above property
\( \displaystyle \int (x + e^x) dx = \int x \; dx + \int e^x \; dx \)
\( \displaystyle \int x \; dx \) is given by 1.3 and \( \displaystyle \int e^x \; dx \)
by 4.1 in table of integral formulas, hence
\[ \int (x + e^x) \; dx = \dfrac{x^2}{2}x + e^x + c \]
Example: Evaluate the integral
\[ \int (2 - 1/x) \; dx \]
Solution:
According to the above property
\( \displaystyle \int (2 - 1/x) dx = \int 2 \; dx - \int (1/x) \; dx \)
\( \int 2 \; dx \) is given by 1.2 and \( \int (1/x) \; dx \) by 1.4 in table of integral formulas, hence
\[ \int (2 - 1/x) \; dx = 2x - \ln |x| + c \]
Example: Evaluate the integral
\[ \int (x^2 - 1)^{20} 2x \; dx \]
Solution:
Let \( u = x^2 - 1\), hence \( du/dx = 2x \) and the given integral can be written as
\( \displaystyle \int(x^2 - 1)^{20} \; 2x \; dx = \int u^{20} (du/dx) dx = \int u^{20} du \)
which evaluates to
\( = \dfrac{u^{21}}{21} + c \)
Substitute back
\( = \dfrac{(x^2 - 1)^{21}}{21} + c \)
Example: Evaluate the integral
\[ \int \; x \; \cos x \; dx \]
Solution:
Let \( f(x) = x \) and \( g ' (x) = \cos x \) which gives
\( f ' (x) = 1 \) and \( g(x) = \sin x \)
From integration by parts formula above,
\( \displaystyle \int \; x \cos x \; dx = x sin x - \int 1 \sin x dx \)
\( = x \sin x + \cos x + c \)
Use the table of integral formulas and the rules above to evaluate the following integrals. [Note that you may need to use more than one of the above rules for one integral].
1. \( \displaystyle \int (1 / 2) \ln \; (x) dx \)
2. \( \displaystyle \int (\sin (x) + x^5 ) \; dx \)
3. \( \displaystyle \int (\sinh (x) - 3) \; dx \)
4. \( \displaystyle \int - x \sin (x) \; dx \)
5. \( \displaystyle \int \sin^{10}(x) \; \cos(x) dx \)