Questions: Evaluate the indefinite integral. (Use symbolic notation and fractions where needed. Use C for the arbitrary constant. Absorb into C as much as possible [ int(7 x+23)^6 d x= ]

$Evaluate the indefinite integral. (Use symbolic notation and fractions where needed. Use C for the arbitrary constant. Absorb into C as much as possible [ int(7 x+23)^6 d x= ]$

Transcript text: Question 7 of 21 Evaluate the indefinite integral. (Use symbolic notation and fractions where needed. Use $C$ for the arbitrary constant. Absorb into $C$ as much as possible \[ \int(7 x+23)^{6} d x= \] $\square$

Solution

Solution Steps

Step 1: Substitution

Let $ u = 7x + 23 $. Then, the differential $ du = 7 \, dx $ implies that $ dx = \frac{1}{7} \, du $.

Step 2: Rewrite the Integral

Substituting $ u $ and $ dx $ into the integral, we have: \[ \int (7x + 23)^6 \, dx = \int u^6 \cdot \frac{1}{7} \, du = \frac{1}{7} \int u^6 \, du \]

Step 3: Integrate

The integral of $ u^6 $ is given by: \[ \int u^6 \, du = \frac{u^7}{7} + C \] Thus, we have: \[ \frac{1}{7} \cdot \left( \frac{u^7}{7} + C \right) = \frac{u^7}{49} + C \]

Step 4: Substitute Back

Substituting back $ u = 7x + 23 $, we get: \[ \frac{(7x + 23)^7}{49} + C \]

Step 5: Final Expression

The final expression for the indefinite integral is: \[ \int (7x + 23)^6 \, dx = \frac{(7x + 23)^7}{49} + C \]