Questions: Use substitution to find the indefinite integral. [ int 8(8 x-3)^7 d x ] [ int 8(8 x-3)^7 d x= ]

Transcript text: Use substitution to find the indefinite integral. \[ \int 8(8 x-3)^{7} d x \] \[ \int 8(8 x-3)^{7} d x= \] $\square$

Solution

Step 1: Substitution

We start by substituting $ u = 8x - 3 $. This gives us $ du = 8 dx $, or equivalently, $ dx = \frac{du}{8} $.

Step 2: Rewrite the Integral

The integral can be rewritten in terms of $ u $: \[ \int 8(8x - 3)^{7} dx = \int 8u^{7} \cdot \frac{du}{8} = \int u^{7} du \]

Step 3: Integrate

Now we integrate $ u^{7} $: \[ \int u^{7} du = \frac{u^{8}}{8} + C \]

Step 4: Substitute Back

Substituting back $ u = 8x - 3 $ into the integral, we have: \[ \frac{(8x - 3)^{8}}{8} + C \]

Thus, the indefinite integral is: \[ \boxed{\frac{(8x - 3)^{8}}{8} + C} \]

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