# Evaluate the integral using the fundamental theorem of calculus.y=4x^t + 5x^-1 + 6sinx.

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We have to find the integral of y=4x^t + 5x^-1 + 6sin x.

I assume t to be a constant.

Int[ y dx]

=> Int[ 4x^t + 5x^-1 + 6sin x dx]

=> Int[4x^t dx] + Int[5x^-1 dx]+ Int [6sin x dx]

=> (4/t+1)*x^(t+ 1) + 5* ln |x| - 6* cos x +C

**The required integral is (4/(t+1))*x^(t +1) + 5*ln |x| - 6* cos x +C**

We'll recall the fundamental Theorem of Calculus:

Int f(x)dx = F(b) - F(a) for x = a to x =b.

Since the limits are not given, we could only calculate the indefinite integral:

Int (4x^t + 5x^-1 + 6sinx)dx

We'll use the additive property of integrals:

Int (4x^t + 5x^-1 + 6sinx)dx = Int 4x^tdx + Int5x^-1dx + Int 6sinxdx

**Int ydx = 4*x^(t+1)/(t+1) + 5 ln|x| - 6cos x + C**