Solve ∫ (from 0 to 1) of cosx-x^2

$$\int _ { 0 } ^ { 1 } \cos x - x ^ { 2 }$$

Evaluate

$$\frac{3\sin(1)-1}{3}\approx 0.508137651$$

Solution Steps

Evaluate the indefinite integral first.

$$\int \cos(x)-x^{2}\mathrm{d}x$$

Integrate the sum term by term.

$$\int \cos(x)\mathrm{d}x+\int -x^{2}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int \cos(x)\mathrm{d}x-\int x^{2}\mathrm{d}x$$

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

$$\sin(x)-\int x^{2}\mathrm{d}x$$

Since \int x^{k}\mathrm{d}x=\frac{x^{k+1}}{k+1} for k\neq -1, replace \int x^{2}\mathrm{d}x with \frac{x^{3}}{3}. Multiply -1 times \frac{x^{3}}{3}.

$$\sin(x)-\frac{x^{3}}{3}$$

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.

$$\sin(1)-\frac{1^{3}}{3}-\left(\sin(0)-\frac{0^{3}}{3}\right)$$

Simplify.

$$\frac{1}{3}\left(3\sin(1)-1\right)$$