What is the integral from 0 to 3 of (2sin(x)-e^x) ?

The integral from 0 to 3 of (2sin(x)-e^x) is 3-2cos(3)-e^3

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integral from 0 to 3 of (2sin(x)-e^x)

\int_{0}^{3} (2 sin (x) - e^{x}) d x

3 - 2 cos (3) - e^{3}

Decimal

- 15.10555 \dots

Solution steps

\int_{0}^{3} 2 sin (x) - e^{x} d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int_{0}^{3} 2 sin (x) d x - \int_{0}^{3} e^{x} d x

\int_{0}^{3} 2 sin (x) d x = 2 (- cos (3) + 1)

\int_{0}^{3} e^{x} d x = e^{3} - 1

= 2 (- cos (3) + 1) - (e^{3} - 1)

Simplify

2 (- cos (3) + 1) - (e^{3} - 1) : 3 - 2 cos (3) - e^{3}

= 3 - 2 cos (3) - e^{3}

What is the integral from 0 to 3 of (2sin(x)-e^x) ?
The integral from 0 to 3 of (2sin(x)-e^x) is 3-2cos(3)-e^3