Answer:
[tex]\int\limits {5^x} \, dx = \frac{5^x}{ln\ x} + c[/tex]
Step-by-step explanation:
Note that the integral of [tex]5^x[/tex] is not [tex]\frac{1}{6}x^6 + c[/tex]
The solution is as follows:
Given
[tex]5^x[/tex]
Required
Integrate
Represent the given expression using integral notation
[tex]\int\limits {5^x} \, dx[/tex]
This question can't be solved directly;
We'll make use of exponential rules which states;
[tex]\int\limits {a^x} \, dx = \frac{a^x}{ln\ x} + c[/tex]
By comparing [tex]\int\limits {5^x} \, dx[/tex] with [tex]\int\limits {a^x} \, dx[/tex];
we can substitute 5 for a;
Hence, the expression [tex]\int\limits {a^x} \, dx = \frac{a^x}{ln\ x} + c[/tex] becomes
[tex]\int\limits {5^x} \, dx = \frac{5^x}{ln\ x} + c[/tex]
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However, the integral of [tex]x^5[/tex] is [tex]\frac{1}{6}x^6 + c[/tex]
This is shown below:
Given that [tex]x^5[/tex]
Applying power rule;
Power rule states that
[tex]\int\limits{x^n}\ dx = \frac{x^{n+1}}{n+1} + c[/tex]
In this case ([tex]x^5[/tex]), n = 5;
So, [tex]\int\limits{x^n}\ dx= \frac{x^{n+1}}{n+1} + c[/tex]
becomes
[tex]\int\limits{x^5}\ dx = \frac{x^{5+1}}{5+1} + c[/tex]
[tex]\int\limits{x^5}\ dx = \frac{x^{6}}{6} + c[/tex]
[tex]\int\limits{x^5}\ dx= \frac{x^{6}}{6} + c[/tex]
