Solve ∫ (from 0 to 0.8767) of (sinx-x^2) wrt x

$$\int _ { 0 } ^ { 0.8767 } ( \sin x - x ^ { 2 } ) d x$$

Evaluate

$$\frac{2326165846337-3000000000000\cos(0.8767)}{3000000000000}\approx 0.135697507$$

Solution Steps

Evaluate the indefinite integral first.

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

Integrate the sum term by term.

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

Factor out the constant in each of the terms.

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

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

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

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}.

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

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.

$$-\cos(0.8767)-\frac{0.8767^{3}}{3}-\left(-\cos(0)-\frac{0^{3}}{3}\right)$$

Simplify.

$$\frac{1}{3000000000000}\left(-3000000000000\cos(0.8767)[phone]\right)$$