Solve ∫ (from y to 298.14) of 26.537+7.6831div1000x-1.172div1000000x^2 wrt x

$$\int_{ y }^{ 298.14 } 26.537+7.6831 \div 1000x-1.172 \div 1000000 { x }^{ 2 } d x$$

Evaluate

$$\frac{293y^{3}}{750000000}-\frac{76831y^{2}}{20000000}-\frac{26537y}{1000}+8242.853779344903744$$

Solution Steps

Expand \frac{7.6831}{1000} by multiplying both numerator and the denominator by 10000.

$$\int _{y}^{298.14}26.537+\frac{76831}{10000000}x-\frac{1.172}{1000000}x^{2}\mathrm{d}x$$

Expand \frac{1.172}{1000000} by multiplying both numerator and the denominator by 1000.

$$\int _{y}^{298.14}26.537+\frac{76831}{10000000}x-\frac{1172}{1000000000}x^{2}\mathrm{d}x$$

Reduce the fraction \frac{1172}{1000000000} to lowest terms by extracting and canceling out 4.

$$\int _{y}^{298.14}26.537+\frac{76831}{10000000}x-\frac{293}{250000000}x^{2}\mathrm{d}x$$

Evaluate the indefinite integral first.

$$\int 26.537+\frac{76831x}{10000000}-\frac{293x^{2}}{250000000}\mathrm{d}x$$

Integrate the sum term by term.

$$\int 26.537\mathrm{d}x+\int \frac{76831x}{10000000}\mathrm{d}x+\int -\frac{293x^{2}}{250000000}\mathrm{d}x$$

Factor out the constant in each of the terms.

$$\int 26.537\mathrm{d}x+\frac{76831\int x\mathrm{d}x}{10000000}-\frac{293\int x^{2}\mathrm{d}x}{250000000}$$

Find the integral of 26.537 using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{26537x}{1000}+\frac{76831\int x\mathrm{d}x}{10000000}-\frac{293\int x^{2}\mathrm{d}x}{250000000}$$

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}. Multiply \frac{76831}{10000000} times \frac{x^{2}}{2}.

$$\frac{26537x}{1000}+\frac{76831x^{2}}{20000000}-\frac{293\int x^{2}\mathrm{d}x}{250000000}$$

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 -\frac{293}{250000000} times \frac{x^{3}}{3}.

$$\frac{26537x}{1000}+\frac{76831x^{2}}{20000000}-\frac{293x^{3}}{750000000}$$

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.

$$26.537\times 298.14+\frac{76831}{20000000}\times 298.14^{2}-\frac{293}{750000000}\times 298.14^{3}-\left(26.537y+\frac{76831}{20000000}y^{2}-\frac{293}{750000000}y^{3}\right)$$

Simplify.

$$\frac{128794590302264121}{15625000000000}-\frac{26537y}{1000}-\frac{76831y^{2}}{20000000}+\frac{293y^{3}}{750000000}$$

Differentiate w.r.t. y

$$\frac{293y^{2}}{250000000}-\frac{76831y}{10000000}-26.537$$