Solve ∫ (from 0 to 4) of (100+50x^2) wrt x

$$\int_{ 0 }^{ 4 } (100+50 { x }^{ 2 } ) d x$$

Evaluate

$$\frac{4400}{3}\approx 1466.666666667$$

Solution Steps

Evaluate the indefinite integral first.

$$\int 100+50x^{2}\mathrm{d}x$$

Integrate the sum term by term.

$$\int 100\mathrm{d}x+\int 50x^{2}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int 100\mathrm{d}x+50\int x^{2}\mathrm{d}x$$

Find the integral of 100 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$100x+50\int x^{2}\mathrm{d}x$$

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

$$100x+\frac{50x^{3}}{3}$$

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.

$$100\times 4+\frac{50}{3}\times 4^{3}-\left(100\times 0+\frac{50}{3}\times 0^{3}\right)$$

Simplify.

$$\frac{4400}{3}$$