if f(x) integral of tan^2x sec^2 xdx and f(0) equals 3, find f(pi/4)
f(x) = ∫tan^2x sec^2x dx = 1/3 tan^3 x + C
f(0) = 3, so C=3 and f(x) = 1/3 tan^3 x + 3
so, f(π/4) = 1/3 + 3 = 10/3
for the second one, you have
y' = 2x-3
so, y = x^2 - 3x + C
since (3,2) is on the graph, y(3) = 2
so, plug that in to find C.
To find the value of f(pi/4), we need to evaluate the integral and then substitute the value pi/4 into the resulting function. Here's how you can do it step by step:
Step 1: Evaluate the integral of f(x)
Since f(x) = ∫(tan^2x)(sec^2x)dx, we can simplify this by using the trigonometric identity: sec^2x = 1 + tan^2x. Substituting this identity into the equation gives:
f(x) = ∫(tan^2x)(1 + tan^2x)dx
Step 2: Simplify the integral
Expanding the expression gives:
f(x) = ∫(tan^4x + tan^2x)dx
Step 3: Perform the integration
The integral of tan^4x can be obtained by using the reduction formula for powers of tangent:
∫tan^4x dx = (1/5)tan^5x - (1/3)tan^3x + C1,
where C1 is the constant of integration.
The integral of tan^2x is straightforward:
∫tan^2x dx = (1/3)tan^3x + C2,
where C2 is the constant of integration.
Step 4: Combine the results
Now we substitute the values from step 3 into the expression for f(x):
f(x) = (1/5)tan^5x - (1/3)tan^3x + (1/3)tan^3x + C2
= (1/5)tan^5x + C2
Step 5: Evaluate f(0)
Given that f(0) = 3, we can substitute x = 0 into the function f(x):
f(0) = (1/5)tan^5(0) + C2
= 0 + C2
= C2
Since f(0) = 3, we can conclude that C2 = 3.
Step 6: Evaluate f(pi/4)
Finally, substitute x = pi/4 into the function f(x) to find the value of f(pi/4):
f(pi/4) = (1/5)tan^5(pi/4) + C2
= (1/5) * 1^5 + 3
= 1/5 + 3
= 1/5 + 15/5
= 16/5
Therefore, f(pi/4) = 16/5.