Evaluate the integral.

∫3−43x4dx

If the integral is divergent, enter ∅.




To evaluate the integral

\[
\int_{3}^{-4} \frac{3}{x^4} \, dx,
\]

we first recognize that the integral has limits of integration from \(3\) to \(-4\). However, typically, the limits of integration should go from a lower value to a higher value. So we will first rewrite the integral by swapping the limits and changing the sign:

\[
\int_{3}^{-4} \frac{3}{x^4} \, dx = -\int_{-4}^{3} \frac{3}{x^4} \, dx.
\]

Now, let's evaluate the integral:

\[
\int \frac{3}{x^4} \, dx.
\]

We can rewrite \(\frac{3}{x^4}\) as \(3x^{-4}\). The antiderivative of \(x^{-4}\) is given by:

\[
\int x^{-4} \, dx = \frac{x^{-3}}{-3} = -\frac{1}{3} x^{-3}.
\]

Therefore, the antiderivative of \(3x^{-4}\) is:

\[
3 \left( -\frac{1}{3} x^{-3} \right) = -x^{-3} = -\frac{1}{x^3}.
\]

So the integral becomes:

\[
-\int_{-4}^{3} \frac{3}{x^4} \, dx = -\left[ -\frac{1}{x^3} \right]_{-4}^{3} = \left[ \frac{1}{x^3} \right]_{-4}^{3}.
\]

Now we evaluate this from \(-4\) to \(3\):

\[
\left[ \frac{1}{x^3} \right]_{-4}^{3} = \frac{1}{3^3} - \frac{1}{(-4)^3} = \frac{1}{27} - \frac{1}{-64}.
\]

Simplifying this, we get:

\[
\frac{1}{27} + \frac{1}{64}.
\]

To combine these fractions, we find a common denominator, which is \(27 \times 64 = 1728\):

\[
\frac{1}{27} = \frac{64}{1728}, \quad \frac{1}{64} = \frac{27}{1728}.
\]

Adding these together:

\[
\frac{64}{1728} + \frac{27}{1728} = \frac{91}{1728}.
\]

Thus, the value of the integral is:

\[
\boxed{\frac{91}{1728}}.
\]

Do you want more details or have any questions?

Here are some more questions you might want to explore:
1. Evaluate \(\int_1^2 \frac{1}{x} \, dx\).
2. Find the derivative of \(f(x) = x^2 \ln x\).
3. Determine the convergence or divergence of \(\sum_{n=1}^\infty \frac{1}{n^3}\).
4. Solve the differential equation \(\frac{dy}{dx} = e^x \cos y\).
5. Compute the integral \(\int_0^{\pi/2} \sin x \, dx\).

Tip: Always ensure the limits of integration are in the correct order; swap them and change the sign if needed to avoid confusion.

Learn how to evaluate the integral ∫[3,-4] 3/x^4 dx step-by-step. This problem involves advanced integral calculus concepts, including the definite integral and antiderivative. Suitable for advanced college-level students.

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