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.
\[
\int x^n \, dx = \frac{x^{n+1}}{n+1} + c
\]
Example: Evaluate the integral
\[ \int x^5 dx \]
Solution:
\[ \int x^5 dx = \dfrac{x^{5 + 1}}{ 5 + 1} + c = \dfrac{x^6}{6} + c \]
\[
\int k f(x) \, dx = k \int f(x) \, dx
\]
Example: Evaluate the integral
\[ \int 5 \sin \; x dx \]
Solution:
According to the above rule
\[ \int 5 \sin (x) dx = 5 \int \sin(x) dx \]
\( \displaystyle \int \sin(x) dx \) is given by 2.1 in table of integral formulas, hence
Hence
\[ \displaystyle \int 5 \sin(x) dx = - 5 \cos x + C \]
\[
\int (f(x) + g(x)) \, dx = \int f(x) \, dx + \int g(x) \, dx
\]
Example: Evaluate the integral
\[ \int (x + e^x) dx \]
Solution:
According to the above property
\[ \displaystyle \int (x + e^x) dx = \int x \; dx + \int e^x \; dx \]
\( \int x \; dx \) is given by 1.3 and \( \displaystyle \int e^x \; dx \)
by 4.1 in table of integral formulas, hence
\[ \int (x + e^x) \; dx = \dfrac{x^2}{2}x + e^x + c \]
\[
\int (f(x) - g(x)) \, dx = \int f(x) \, dx - \int g(x) \, dx
\]
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
\[ \int (2 - 1/x) \; dx = 2x - \ln |x| + c \]
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. \( \displaystyle \int \sin^{10}(x) \; \cos(x) dx \)
Use rule 3 (integral of a sum) to obtain:
\[ \int (\sin x + x^5) \, dx = \int \sin x \, dx + \int x^5 \, dx \]We use formula 2.1 in the table of integral formulas to evaluate \( \int \sin x \, dx \) and rule 1 above to evaluate \( \int x^5 \, dx \). Hence:
\[ \int (\sin x + x^5) \, dx = -\cos x + \frac{x^6}{6} + c \] 3.Use rule 4 (integral of a difference) to obtain:
\[ \int (\sinh x - 3) \, dx = \int \sinh x \, dx - \int 3 \, dx \]We use formula 7.1 in the table of integral formulas to evaluate \( \int \sinh x \, dx \) and the integral of the constant 3 to obtain:
\[ \int (\sinh x - 3) \, dx = \cosh x - 3x + c \] 4.The integrand is the product of two functions \( x \) and \( \sin x \). We use integration by parts (rule 6) as follows:
Let \( f(x) = x \), \( g'(x) = \sin x \), and hence \( g(x) = -\cos x \).
Then:
\[ \int x \sin x \, dx = f(x) g(x) - \int f'(x) g(x) \, dx = -x \cos x + \int \cos x \, dx \]Using formula 2.2 in the table of integral formulas to evaluate \( \int \cos x \, dx \), we obtain:
\[ \int x \sin x \, dx = - x \cos x + \sin x + c \] 5.Let \( u = \sin x \) and therefore \( du = \cos x \, dx \). Hence the given integral can be written as:
\[ \int \sin^{10} x \cos x \, dx = \int u^{10} \, du \]Use rule 1 to write:
\[ \int u^{10} \, du = \frac{u^{11}}{11} + c \]Substitute \( u = \sin x \) to obtain:
\[ \int \sin^{10} x \cos x \, dx = \frac{1}{11} \sin^{11} x + c \]