To determine

To find:

Equation of the tangent line

Answer

y=-xπ+1

Explanation

1) Concept:

i) Fundamental Theorem of calculus. If f(x) is continuous on [a, b] and

gx=∫axf(t) dt

then g'x=f(x)

2) Formula:

i) Slope point form:

y-y1=m ·(x-x1)

3) Given:

Fx=∫πxcos⁡tt dt

4) Calculation:The slope of tangent to the curve at x is given by derivative at x.

Here

Fx=∫πxcos⁡tt dt

By using Fundamental Theorem of Calculus,

F’x= cos⁡xx

Given that, x=π,

Substitute value of x in F’(x)

F’(π)=cos⁡ππ

Since, cos⁡π=-1

F’π=-1π

Therefore, slope of tangent line at x=π is

-1π At x=π

Fx=∫πxcos⁡tt dt= ∫ππcos⁡xx dx=0 since upper and lower limits of integral are same.

Therefore, Fπ=0

Use point slope form to find equation of tangent line,

y-y1=m ·(x-x1)

Now, m=-1π and x1=π ,y1=0

Substituting the value,

y-0=-1π ·(x-π)

y=-1π ·(x-π)

Applying distributive property,

y=-xπ+1

This is the equation of tangent line to the curve, y=Fx at the point π,0.

Conclusion:

Therefore, the equation of tangent line is

y=-xπ+1