Find an antiderivative P(t) of p(t)=t^7-(t^5/5)-t
(1/8) t^8 - (1/30)t^6 - (1/2)t^2
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P(t) =(18)t^8 - (1/30)t^6 - (1/2)t^2 + c
To find an antiderivative P(t) of the function p(t) = t^7 - (t^5/5) - t, we can use the power rule for integration.
The power rule states that if we have a term of the form x^n, where n is any real number except -1, the antiderivative is (1/(n+1)) * x^(n+1) + C, where C is the constant of integration.
Let's break down the function p(t) into individual terms and find their antiderivatives:
1. Antiderivative of t^7: Using the power rule, we add 1 to the exponent (7 + 1) and divide the term by the new exponent, which gives us (1/8) * t^8.
2. Antiderivative of (t^5)/5: Here, we have a constant factor of 1/5 multiplied by t^5. Applying the power rule, we divide the term by the new exponent (5 + 1 = 6) and also divide the constant by the new exponent, resulting in (1/30) * t^6.
3. Antiderivative of -t: In this case, we have a constant factor of -1 multiplied by t. Integrating -t gives us -(1/2) * t^2, applying the power rule.
Now, we can sum up the antiderivatives of each term to find the overall antiderivative P(t):
P(t) = (1/8) * t^8 + (1/30) * t^6 - (1/2) * t + C
Therefore, the antiderivative of p(t) = t^7 - (t^5/5) - t is P(t) = (1/8) * t^8 + (1/30) * t^6 - (1/2) * t + C, where C is the constant of integration.