Could someone answer this question. im stuck on it. Id appreciate it.
Use substitution to find the indefinite integral.
�çp(p+3)^5 dp
Use C as the arbitrary constant.
To find the indefinite integral of the expression �çp(p+3)^5 dp , we can use the method of substitution.
Step 1: Choose a suitable substitution.
In this case, let's substitute u = p + 3.
Step 2: Calculate the derivative of the substitution.
Differentiating u = p + 3 with respect to p gives du/dp = 1.
Step 3: Rearrange the substitution equation to solve for dp.
Since u = p + 3, we can rewrite it as p = u - 3.
Step 4: Substitute the value of dp in terms of du into the original integral.
Using the equation from Step 3, we have p = u - 3. Therefore, dp = du.
Substituting these values into the original integral, we have:
�çp(p+3)^5 dp = �ç(u-3)u^5 du
Step 5: Simplify the new integral expression.
Expanding (u-3)u^5 gives: �ç(u^6 - 3u^5) du
Step 6: Integrate the simplified expression.
Integrating each term individually, we get:
∫(u^6 - 3u^5) du = ∫u^6 du - ∫3u^5 du
Integrating each term using the power rule, we have:
(1/7)u^7 - (3/6)u^6 + C
Finally, substitute back the original variable p to get the answer in terms of p:
(1/7)(p+3)^7 - (3/6)(p+3)^6 + C
Thus, the indefinite integral of �çp(p+3)^5 dp is (1/7)(p+3)^7 - (3/6)(p+3)^6 + C, where C is the arbitrary constant.