Consider the function $f(t)=8 \sec ^2(t)-9 t^3$. Let $F(t)$ be the antiderivative of $f(t)$ with $F(0)=0$. Then $F(t)=$ $\square$

The antiderivative of the function f ( t ) = 8 sec 2 ( t ) − 9 t 3 with the condition F ( 0 ) = 0 is F ( t ) = 8 tan ( t ) − 4 9 ​ t 4 . This solution is derived by integrating each term and applying the initial condition to find the constant. The final answer represents the integral of the given function.
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Answered by Anonymous | 2025-07-04

Find the antiderivative of 8 sec 2 ( t ) , which is 8 tan ( t ) .
Find the antiderivative of − 9 t 3 , which is − 4 9 ​ t 4 .
Combine the antiderivatives: F ( t ) = 8 tan ( t ) − 4 9 ​ t 4 + C .
Use the initial condition F ( 0 ) = 0 to find C = 0 , so F ( t ) = 8 tan ( t ) − 4 9 ​ t 4 . The final answer is 8 tan ( t ) − 4 9 ​ t 4 ​ .

Explanation

Problem Analysis We are given the function f ( t ) = 8 sec 2 ( t ) − 9 t 3 and asked to find its antiderivative F ( t ) such that F ( 0 ) = 0 . This is an initial value problem, where we need to find the indefinite integral of f ( t ) and then use the initial condition to determine the constant of integration.

Antiderivative of 8 sec 2 ( t ) First, we find the antiderivative of 8 sec 2 ( t ) . Recall that the derivative of tan ( t ) is sec 2 ( t ) . Therefore, the antiderivative of 8 sec 2 ( t ) is 8 tan ( t ) .

Antiderivative of − 9 t 3 Next, we find the antiderivative of − 9 t 3 . Using the power rule for integration, we have ∫ − 9 t 3 d t = − 9 ∫ t 3 d t = − 9 ⋅ 3 + 1 t 3 + 1 ​ + C = − 9 ⋅ 4 t 4 ​ + C = − 4 9 ​ t 4 + C

Combining Antiderivatives Combining these results, we have F ( t ) = ∫ ( 8 sec 2 ( t ) − 9 t 3 ) d t = 8 tan ( t ) − 4 9 ​ t 4 + C

Finding the Constant of Integration Now we use the initial condition F ( 0 ) = 0 to find the constant C :
F ( 0 ) = 8 tan ( 0 ) − 4 9 ​ ( 0 ) 4 + C = 0 Since tan ( 0 ) = 0 , we have 0 − 0 + C = 0 Thus, C = 0 .

Final Answer Therefore, the antiderivative F ( t ) is F ( t ) = 8 tan ( t ) − 4 9 ​ t 4


Examples
In physics, if f ( t ) represents the acceleration of an object at time t , then F ( t ) represents the object's velocity at time t . The initial condition F ( 0 ) = 0 would mean the object starts at rest. Determining the velocity function from the acceleration is a common application of finding antiderivatives.

Answered by GinnyAnswer | 2025-07-04