Contents
Our article today is a diverse collection of indefinite integral problems that include many solved, supportive problems.
However, before reading the following problems, please review the indefinite integral rules lesson. Make sure to memorize and master them.
Indefinite Integral Problems
Problem 1 (Power Function and Constant Function Integration)
Find the integral of the following function:
\[ f\left( x \right) = {x^4} – 5x + 9\]
The term x^4 and the term -5x are solved according to the power rule
\[ ∫ xⁿ dx = \frac{x^{n+1}}{n+1} + C\]
And the third term (9) is solved according to the constant function rule:
\[∫ a dx = ax + C , a : constant\]
Thus, the solution is:
\[F\left( x \right) = \frac{1}{5}{x^5} – \frac{5}{2}{x^2} + 9x + c,\,\,\hspace{0.25in}c{\mbox{ is a constant}}\]
Problem 2
Find the result of the following integral:
\[\displaystyle \int{{5{t^3} – 7{t^{ – 6}} + 4\,dt}}\]
This is a standard exercise where we use the first and second rules. The constant and the power. We add 1 to the exponent and divide by the new exponent. We must pay close attention to the signs, as this is where students often lose marks.
\[\begin{align}\int{{5{t^3} – 7{t^{ – 6}} + 4\,dt}} & = 5\left( {\frac{1}{4}} \right){t^4} – 7\left( {\frac{1}{{ – 5}}} \right){t^{ – 5}} + 4t + c\\& = \frac{5}{4}{t^4} + \frac{7}{5}{t^{ – 5}} + 4t + c\end{align}\]
Problem 3
Find the result of the following integral:
\[\displaystyle \int{{3\sqrt[4]{{{x^3}}} + \frac{7}{{{x^5}}} + \frac{1}{{6\sqrt x }}\,dx}}\]
At first glance, it looks like a difficult integral. But there’s nothing new; all we have to do is follow one of the previous rules, which is the power rule. All we need to know is that the square root can be written as a power of (1/2). That is: \(\sqrt{x} = x^\frac{1}{2}\)
And we pay attention to the degree of the root; the degree of the square root is 2, and its symbol is \(\sqrt{}\), while this root, for example, \(\sqrt[4]{x}\), has a degree of 4 and can be written as: \(x^{\frac{1}{4}}\)
And if the root is in the denominator, it can be moved to the numerator by inverting the sign of the exponent, i.e.: \(\frac{1}{\sqrt{x}} = x^{-\frac{1}{2}}\)
Based on the above, we conclude that all terms can be solved with the power rule.
\[\begin{align}\int{{3\sqrt[4]{{{x^3}}} + \frac{7}{{{x^5}}} + \frac{1}{{6\sqrt x }}\,dx}} & = \int{{3{x^{\frac{3}{4}}} + 7{x^{ – 5}} + \frac{1}{6}{x^{ – \,\,\frac{1}{2}}}\,dx}}\\ & = 3\frac{1}{{{}^{7}/{}_{4}}}{x^{\frac{7}{4}}} – \frac{7}{4}{x^{ – 4}} + \frac{1}{6}\left( {\frac{1}{{{}^{1}/{}_{2}}}} \right){x^{\frac{1}{2}}} + c\\ & = \frac{{12}}{7}{x^{\frac{7}{4}}} – \frac{7}{4}{x^{ – 4}} + \frac{1}{3}{x^{\frac{1}{2}}} + c\end{align}\]
Problem 4
Solve the following integral problem: \(\displaystyle \int{{dw}}\)
The problem seems confusing at first. However, it is very easy and can be solved using the constant function rule. The integration is with respect to the variable w.
The function can be written as follows:
\[\displaystyle \int{{1 \ dw}}\]
This means we are integrating the number 1! Therefore, the answer is:
\[w+c\]
Problem 5 (Logarithmic Function)
Find the result of the following indefinite integral:
\[\displaystyle \int{{\frac{{5{x^{11}} – 4{x^4} – 7{x^2}}}{{{x^3}}}\,dx}}\]
Solution: We divide each term in the numerator by the x3 in the denominator, so we have:
\[\int{{\frac{5x^{11}}{x^3}- \frac{4x^{4}}{x^3}-\frac{7x^{2}}{x^3}\,dx}}\]
Now we simplify:
\[\int{5x^8-4x-\frac{7}{x}\,dx}\]
The first and second terms can be integrated with the power rule. But the last one cannot. Let’s try:
\[\int{7x^{-1}\, dx} =\frac{ 7x^{-1+1}}{-1+1}\]
As we can see, division by zero is not possible (an indeterminate form), so we cannot integrate this term using the power rule. We resort to the logarithmic rule in integration, which is:
\[\int{{\frac{1}{x}\,dx}} = \int{{{x^{ – 1}}\,dx}} = \ln \left| x \right| + c\]
Thus, the integral of the last term is:
\[\int{{-\frac{{7}}{x}\,dx}} = -7\int{{\frac{1}{x}\,dx}} = -7\ln \left| x \right| + c\]
And the final result of the integral is:
\[\frac{5}{9}{x^9} – 2{x^2} – 7\ln \left| x \right| + c\]
Problem 6 (Power Functions and Trigonometric Functions)
Find the result of the following integral:
\[\int{{4{{\bf{e}}^x} – 3\sin x – 6{{\sec }^2}x\,dx}}\]
Solution: This integral is simple and is a direct application of the integration rule for the natural exponential function and trigonometric functions:
\[= 4{{\bf{e}}^x} + 3\cos x – 6\tan x + c\]
Write to us in the comments to receive a PDF file on indefinite integral problems.
