Solve ∫ (from 0 to 1) of (sinx-cosx+e^x) wrt x=

$$\int _ { 0 } ^ { 1 } ( \sin x - \cos x + e ^ { x } ) d x =$$

Evaluate

$$e-\sin(1)-\cos(1)\approx 1.336508538$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \sin(x)-\cos(x)+e^{x}\mathrm{d}x$$

Integrate the sum term by term.

$$\int \sin(x)\mathrm{d}x+\int -\cos(x)\mathrm{d}x+\int e^{x}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int \sin(x)\mathrm{d}x-\int \cos(x)\mathrm{d}x+\int e^{x}\mathrm{d}x$$

Use \int \sin(x)\mathrm{d}x=-\cos(x) from the table of common integrals to obtain the result.

$$-\cos(x)-\int \cos(x)\mathrm{d}x+\int e^{x}\mathrm{d}x$$

Use \int \cos(x)\mathrm{d}x=\sin(x) from the table of common integrals to obtain the result.

$$-\cos(x)-\sin(x)+\int e^{x}\mathrm{d}x$$

Use \int e^{x}\mathrm{d}x=e^{x} from the table of common integrals to obtain the result.

$$-\cos(x)-\sin(x)+e^{x}$$

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.

$$-\cos(1)-\sin(1)+e^{1}-\left(-\cos(0)-\sin(0)+e^{0}\right)$$

Simplify.

$$e-\cos(1)-\sin(1)$$