Solve ∫ (from -2 to 7) of 4x^2+sin(x) wrt x

$$\int_{ -2 }^{ 7 } 4 { x }^{ 2 } + \sin ( x ) d x$$

Evaluate

$$\cos(2)+468-\cos(7)\approx 466.829950909$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 4x^{2}+\sin(x)\mathrm{d}x$$

Integrate the sum term by term.

$$\int 4x^{2}\mathrm{d}x+\int \sin(x)\mathrm{d}x$$

Factor out the constant in each of the terms.

$$4\int x^{2}\mathrm{d}x+\int \sin(x)\mathrm{d}x$$

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 4 times \frac{x^{3}}{3}.

$$\frac{4x^{3}}{3}+\int \sin(x)\mathrm{d}x$$

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

$$\frac{4x^{3}}{3}-\cos(x)$$

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.

$$\frac{4}{3}\times 7^{3}-\cos(7)-\left(\frac{4}{3}\left(-2\right)^{3}-\cos(-2)\right)$$

Simplify.

$$468-\cos(7)+\cos(2)$$