Question

Evaluate the indefinite integral. (Use C for the constant of integration.)
x7 sin(x8) dx

Answer 1

To evaluate the indefinite integral \(\int x^7 \sin(x^8) \, dx\), we can use the method of substitution.

Let:
\[
u = x^8
\]
Then, the differential \(du\) is given by:
\[
du = 8x^7 \, dx \quad \Rightarrow \quad dx = \frac{du}{8x^7}
\]
Now, substitute \(u\) in the integral. The \(x^7\) in \(dx\) will cancel out:
\[
\int x^7 \sin(x^8) \, dx = \int x^7 \sin(u) \cdot \frac{du}{8x^7} = \frac{1}{8} \int \sin(u) \, du
\]

The integral of \(\sin(u)\) is:
\[
\int \sin(u) \, du = -\cos(u)
\]

Thus, we have:
\[
\frac{1}{8} \int \sin(u) \, du = \frac{1}{8} (-\cos(u)) = -\frac{1}{8} \cos(u) + C
\]

Now substitute back \(u = x^8\):
\[
-\frac{1}{8} \cos(x^8) + C
\]

So, the final answer is:
\[
\int x^7 \sin(x^8) \, dx = -\frac{1}{8} \cos(x^8) + C
\]