Master the Power Rule for Integrating ∫a^x dx: A Substitution Technique

∫a^x dx

To evaluate the integral ∫a^x dx, we can use the power rule for integration

To evaluate the integral ∫a^x dx, we can use the power rule for integration. However, we need to be careful when dealing with a variable base raised to the power of x.

The power rule states that if we have an integral of the form ∫x^n dx, where n is any real number except -1, the result is (1/(n+1)) * x^(n+1) + C, where C is the constant of integration.

In our case, the base a is a constant, not a variable. Therefore, we cannot directly apply the power rule. Instead, we need to use a substitution technique to convert the integral to a form where we can apply the power rule.

Let’s make the substitution u = a^x. Now, we need to find the corresponding derivative of u with respect to x.

Taking the natural logarithm of both sides of the equation u = a^x, we have ln(u) = ln(a^x). Using the logarithmic property, we can rewrite this as ln(u) = x * ln(a).

Differentiating both sides with respect to x, we get (1/u) * du/dx = ln(a). Now, solving for du/dx, we have du/dx = u * ln(a).

So, du/dx = a^x * ln(a).

Now, let’s go back to the original integral and substitute u = a^x and du/dx = a^x * ln(a):

∫a^x dx = ∫(du/dx) dx.

Since du/dx = a^x * ln(a), we can rewrite the integral as:

∫a^x dx = ∫du.

Notice that when we made the substitution, the original integration variable x transformed into the new variable u. This means that the new integral with respect to u is simply u + C, where C is the constant of integration.

Therefore, the final result is:

∫a^x dx = u + C = a^x + C.

So, the integral of a^x with respect to x is a^x + C, where C is the constant of integration.

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