Solve ∫ (from 1 to 4) of 1+e^-2y

$$\int _ { 1 } ^ { 4 } 1 + e ^ { - 2 y }$$

Evaluate

$$\frac{3}{e^{2y}}+3$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 1+\frac{1}{e^{2y}}\mathrm{d}x$$

Find the integral of 1+\frac{1}{e^{2y}} using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\left(1+\frac{1}{e^{2y}}\right)x$$

The definite integral is the antiderivative of the expression evaluated at the upper limit of integration minus the antiderivative evaluated at the lower limit of integration.

$$\left(1+e^{-2y}\right)\times 4-\left(1+e^{-2y}\right)$$

Simplify.

$$3+\frac{3}{e^{2y}}$$

Differentiate w.r.t. y

$$-\frac{6}{e^{2y}}$$