# Recent questions in Applications of integrals

Applications of integrals

### Explain why each of the following integrals is improper. (a) $$\int_6^7 \frac{x}{x-6}dx$$ -Since the integral has an infinite interval of integration, it is a Type 1 improper integral. -Since the integral has an infinite discontinuity, it is a Type 2 improper integral. -The integral is a proper integral. (b)$$\int_0^{\infty} \frac{1}{1+x^3}dx$$ Since the integral has an infinite interval of integration, it is a Type 1 improper integral. Since the integral has an infinite discontinuity, it is a Type 2 improper integral. The integral is a proper integral. (c) $$\int_{-\infty}^{\infty}x^2 e^{-x^2}dx$$ -Since the integral has an infinite interval of integration, it is a Type 1 improper integral. -Since the integral has an infinite discontinuity, it is a Type 2 improper integral. -The integral is a proper integral. d)$$\int_0^{\frac{\pi}{4}} \cot x dx$$ -Since the integral has an infinite interval of integration, it is a Type 1 improper integral. -Since the integral has an infinite discontinuity, it is a Type 2 improper integral. -The integral is a proper integral.

Applications of integrals

### Evaluate this indefinite integral: $$\int(3x+5)dx$$

Applications of integrals

### Given $$\int_{2}^{5}f(x)dx=17$$ and $$\int_{2}^{5}g(x)dx=-2$$, evaluate the following. (a)$$\int_{2}^{5}[f(x)+g(x)]dx$$ (b)$$\int_{2}^{5}[g(x)-f(x)]dx$$ (c)$$\int_{2}^{5}2g(x)dx$$ (d)$$\int_{2}^{5}3f(x)dx$$

Applications of integrals

### Evaluate each of the following integrals. $$\int_{0}^{2}(x^{2}+2x-3)^{3}(4x+4)dx$$

Applications of integrals

### Find formulas for the functions represented by the integrals. $$\int_{-\frac{\pi}{4}}^{x}\sec^{2}0d0$$

Applications of integrals

### Find the length of the curve. $$r(t)=<2t, t^2, \frac{1}{3}t^3>$$ $$0\leq t\leq1$$

Applications of integrals

### What is the integral of the constant function $$f (x, y) = 5$$ over the rectangle $$[-2,3]\times[2,4]$$?

Applications of integrals

### Suppose $$\int_{-2}^{2}f(x)dx=4,\int_{2}^{5}f(x)dx=3, \int_{-2}^{5}g(x)dx=2$$. Is the following statement true? $$\int_{-2}^{5}(f(x)+g(x))=9$$

Applications of integrals

### Evaluate the following integral: $$\int\frac{vdv}{6v^{2}-1}$$

Applications of integrals

### Evaluate the following integral: $$\int \frac{x+3}{x-1}dx$$

Applications of integrals

### Evaluate the ff, improper integrals. $$\int_{1}^{\infty}\frac{1}{x^{3}}dx$$

Applications of integrals

### Evaluate the following integral. $$\int \frac{3x^{2}+\sqrt{x}}{\sqrt{x}}dx$$

Applications of integrals

### Evaluate the iterated integral $$\int_{-1}^{2}\int_{0}^{\frac{\pi}{2}}$$

Applications of integrals

### Evaluate the following integral. $$\int 2x^{3}+3x-2dx$$

Applications of integrals

### use the table of integrals to evaluate the following integral. $$\int 3x \sqrt{6x-x^2}dx$$

Applications of integrals

### Find the indefinite integral $$\int \ln(\frac{x}{3})dx$$ (a) using a table of integrals and (b) using the Integration by parts method.

Applications of integrals

### Use a change of variables to evaluate the following integral. $$\int-(\cos^{7}x-5\cos^{5}x-\cos x)\sin x dx$$

Applications of integrals

### Evaluate the ff, improper integrals. $$\int_{-2}^{\infty}\sin x dx$$

Applications of integrals