Solve ∫ (from - 1 to 1) of (x+k)

$$\int _ { - 1 } ^ { 1 } ( x + k )$$

Evaluate

$$2k$$

Solution Steps

Evaluate the indefinite integral first.

$$\int x+k\mathrm{d}x$$

Integrate the sum term by term.

$$\int x\mathrm{d}x+\int k\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x\mathrm{d}x with \frac{x^{2}}{2}.

$$\frac{x^{2}}{2}+\int k\mathrm{d}x$$

Find the integral of k using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{x^{2}}{2}+kx$$

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.

$$\frac{1^{2}}{2}+k\times 1-\left(\frac{\left(-1\right)^{2}}{2}+k\left(-1\right)\right)$$

Simplify.

$$2k$$

Differentiate w.r.t. k

$$2$$