## Thursday, September 29, 2022

## Wednesday, September 28, 2022

## $\int \frac{x}{x-1}dx$ integral | What is the integrate of x/x-1 ?

## Monday, September 26, 2022

## $\sqrt{1+\sqrt{1+\sqrt{x}}}$ derivative | What is the derivative of $\frac{d}{dx}\sqrt{1+\sqrt{1+\sqrt{x}}}$?

Hello everyone. This lesson we will tell the derivative of $\sqrt{1+\sqrt{1+\sqrt{x}}}$.

## What is the derivative of $\sqrt{1+\sqrt{1+\sqrt{x}}}$?

**Compute the derivative of $\sqrt{1+\sqrt{1+\sqrt{x}}}$.**

Here we have a more complicated chain of compositions, so we use the chain rule twice. At the outermost “layer” we have the function $g(x)=1+\sqrt{1+\sqrt{x}}$ plugged into $f(x)=\sqrt{x}$, so applying the chain rule once gives

$\frac{d}{dx}\sqrt{1+\sqrt{1+\sqrt{x}}}=\frac{1}{2}(1+\sqrt{1+\sqrt{x}})^{\frac{-1}{2}}.\frac{d}{dx}(1+\sqrt{1+\sqrt{x}})$

**Not we need the derivative**of $\sqrt{1+\sqrt{x}}$. Using the chain rule again:

$\frac{d}{dx}\sqrt{1+\sqrt{x}}=\frac{1}{2}(1+\sqrt{x})^{\frac{-1}{2}}.\frac{1}{2}x^{\frac{-1}{2}}$

**So the original derivative is**

$\frac{d}{dx}\sqrt{1+\sqrt{1+\sqrt{x}}}=\frac{1}{2}(1+\sqrt{1+\sqrt{x}})^{\frac{-1}{2}}.\frac{1}{2}(1+\sqrt{x})^{\frac{-1}{2}}.\frac{1}{2}x^{\frac{-1}{2}}$

=$\frac{1}{8\sqrt{x}.\sqrt{1+\sqrt{x}}.\sqrt{1+\sqrt{1+\sqrt{x}}}}$

Using the chain rule, the power rule, and the product rule, it is possible to avıid using the quotient rule entirely.

## Product Rule for Derivative | How to take the multiplication derivative?

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categories: multiplication derivative, product rule, product rule of derivative

categories: multiplication derivative, product rule, product rule of derivative

Hello everyone. This lesson we will tell the product rule for derivative.

## Product Rule for Derivative

product rule of derivative |

Let f(x) and g(x) be two functions. Then the derivate of the product

**$(f(x).g(x))'=f'(x)g(x)+f(x)g'(x)$**

We must follow this rule religiously and not succumb to the temptation of writing $(f(x)g(x))'=f'(x)g'(x)$ ; a faulty statement.

**Example:**

$(x^3.e^x)'=(x^3)'.e^x+x^3(e^x)'$

=$3x^2e^x+x^3e^x$

## Saturday, September 24, 2022

## Quotient Rule for Derivative | How to take the derivative of the quotient?

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categories: derivative of the quotient, quotient rule, quotient rule for derivative

categories: derivative of the quotient, quotient rule, quotient rule for derivative

Hello everyone. This lesson we will tell the quotient rule for derivative.

## Quotient Rule for Derivative

quotient rule derivative |

Let ݂f(x) and ݃g(x) be two functions. Then the derivative of the quotient:

**$(\frac{f(x)}{g(x)})'=\frac{f'(x).g(x)-f(x).g'(x)}{[g(x)]^2}$**

This is how the derivative of the quotient is taken. Now let's reinforce the issue with an example.

**Example:**

What is the derivative of $(\frac{x^3}{e^x})'$

$(\frac{x^3}{e^x})'=\frac{(x^3)'.e^x-x^3.(e^x)'}{(e^x)^2}$

$=\frac{3x^2e^x-x^3e^x}{(e^x)^2}$

$=\frac{x^2e^2.(3-x)}{(e^x)^2}$

**$=\frac{x^2(3-x)}{e^x}$**

## Graph of cotangent | What is the cot(x) graph drawing?

Hello everyone dear friends. In this lesson, we will share with you what the cotangent(x) graph is.

## What is the graph of cotx?

**The plot of cot(x) is as follows:**

cotangent graph |

graph of cot(x) |

Both graphs are graphs of cotangent(x). You can use both. Thank you so much. We wish you success in your studies.

## Thursday, September 22, 2022

## $e^{1-x}dx$ integral | What is integral of e^1-x ?

## Graph of tangent | What is the tan(x) graph drawing?

## Wednesday, September 21, 2022

## $\int \frac{x^2+1}{x^2-1}dx=?$ | What is integral of (x^2+1)/(x^2-1)

You can reach integral of (x^2+1)/(x^2-1) answer on this page.

## What is integral of $\int \frac{x^2+1}{x^2-1}dx$?

integral of (x^2+1)/(x^2-1) |

**Solution:**

**Performing polynomial long division, we have that:**

$\int \frac{x^2+1}{x^2-1}dx=\int (1+\frac{2}{x^2-1})dx$

$=\int dx + \int \frac{2}{x^2-1}dx$

$=x+\int \frac{2}{x^2-1}dx$

**Using partial fraction on the remaining integral, we get:**

$\frac{2}{x^2-1}=\frac{A}{x-1}+\frac{B}{x+1}=\frac{A(x+1)+B(x-1)}{(x+1)(x-1)}=\frac{(A+B)x+(A-B)}{x^2-1}$

Thus, A + B = 0 and A − B = 2. Adding the two equations together yields 2.A = 2, that is, A = 1, and B = − 1. So, we have that:

$\int \frac{2}{x^2-1}dx=\int \frac{1}{x-1}dx-\int \frac{1}{x+1}dx$

**Therefore,**

$\int \frac{x^2+1}{x^2-1}dx=x+\int \frac{2}{x^2-1}dx$

$=x+\int \frac{1}{x-1}dx-\int \frac{1}{x+1}dx$

$=x+ln|x-1|-ln|x+1|+C$

**Answer :**

**$\int \frac{x^2+1}{x^2-1}dx=x+ln|x-1|-ln|x+1|+C$**

## Tuesday, September 20, 2022

## $\int \frac{1}{x\sqrt{x}}dx$ | What is integral of 1/x√x

## $\int sin^5{x}dx=?$ | What is integral of sinx^5

You can reach integral of sinx^5 answer on this page.

**What is integral of ∫sinx^5dx?**

integrate of sinx^5 |

Evaluate $\int sin^5{x}dx$.

**Rewrite the function:**

$\int sin^5{x}.dx=\int sinx.sin^4x.dx=\int sinx(sin^2x)^2.dx=\int sinx(1-cos^2x)^2.dx$

**Now use u=cosx, du=-sinxdx:**

$\int sinx(1-cos^2x)^2.dx=\int -(1-u^2)^2du=\int -(1-2u^2+u^4)du$

=$-u+\frac{2}{3}u^3-\frac{1}{5}u^5+C$

=$-cosx+\frac{2}{3}cos^3x-\frac{1}{5}cos^5x+C$

**Answer:**

**$\int sin^5{x}.dx=-cosx+\frac{2}{3}cos^3x-\frac{1}{5}cos^5x+C$**

## Monday, September 19, 2022

## $\int \frac{lnx^2}{x}.dx=?$ | What is integral of lnx^2/x?

You can reach integral of (lnx^2)/x answer on this page.

## What is integral of ∫lnx^2/xdx?

**Question:**

$\int \frac{lnx^2}{x}.dx=?$

lnx^2/x integrate |

**Solution:**

Substituting u = lnx and $du = \frac{1}{x}dx$ , you get

$\int \frac{lnx^2}{x}dx=\int \frac{2lnx}{x}dx=2.\int u.du=2.\frac{1}{2}u^2+C = (lnx)^2+C$

**$\int \frac{lnx^2}{x}.dx=(lnx)^2+C$**

## Sunday, September 18, 2022

## $x.e^{x^2}$ integral | What is integral of x.e^(x^2) ?

## Graph of sin | What is the sin(x) graph drawing?

Hello everyone dear friends. In this lesson, we will share with you what the sin graph is.

**What is the graph of sin(x)?**

**The plot of sin(x) is as follows:**

sin graph |

sinx graph |

sinx graph |

Here are 3 different sinx charts.

All three graphs are sinx's graphs. You can use all three. Thank you so much. We wish you success in your work.

## Saturday, September 17, 2022

## Graph of cos | What is the cos(x) graph drawing?

## How to find the slope of a graph? (Examples and answers)

Hello everyone, in this article, we will tell you how to find the slope of a graph and how to calculate it with examples. We wish you a good work ahead...

**How to find the slope of a graph?**

The slope of a line is determined by the ratio $\frac{change in y}{change in x}$ between any two points that lie on the line.

The slope is the constant rate of change of a line.

**Example:**

Use a graph to determine the slope of a line.

Slope of graph = -1/2 |

**Step 1:**Identify two points on the line. In this case, use (0, 2) and (2, 1).

**Step 2:**Calculate the vertical change from one point to the next. In this case, you must count down 1 space to move from the point (0, 2) to the point (2, 1).

**Step 3:**Calculate the horizontal change from one point to the next. In this case, you must count right 2 spaces to move from the point (0, 2) to the point (2, 1).

**Step 4:**Write the ratio showing $\frac{vertical change}{horizontal change}$ in simplest form.

In this case, the slope is represented by the ratio $\frac{-1}{2}$ , or $-\frac{1}{2}$ .

**Solution:**The slope is negative because the line falls from left to right.

**Answer = -1/2**

**Practice:**

*Source : collegeboard.org*

## Thursday, September 15, 2022

## $\int \frac{1}{x.lnx}.dx=?$ | What is integral of 1/x.lnx?

You can reach integral of 1/x.lnx answer on this page.

**What is integral of ∫1/x.lnx.dx?**

**Question:**

$\int \frac{1}{x.lnx}.dx=?$

**Solution:**

Substituting $u = lnx$ and $du = \frac{1}{x}dx$, you get

$\int \frac{1}{x.lnx}.dx=\int \frac{1}{u}.du=ln|u|+C = ln|lnx| + C$

**$\int \frac{1}{x.lnx}.dx=ln|lnx| + C$**

## Wednesday, September 14, 2022

## Homework: In problems 1 through 20 find the area of the region R

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categories: area of region homeworks solutions, area of region problems, find area of region, integral answer

categories: area of region homeworks solutions, area of region problems, find area of region, integral answer

**Questions and answers of find area of region problems.**

**1-)**R is the triangle bounded by the line y = 4 − 3x and the coordinate axes.

**Answer =**$\frac{8}{3}$

**2-)**R is the rectangle with vertices (1, 0), (−2, 0), (−2, 5) and (1, 5).

**Answer =**15

**3-)**R is the trapezoid bounded by the lines y = x + 6 and x = 2 and the coordinate axes.

**Answer =**14

**4-)**R is the region bounded by the curve $y = \sqrt{x}$ , the line x = 4, and the x axis.

**Answer =**$\frac{16}{3}$

**5-)**R is the region bounded by the curve $y = 4x^3$ , the line x = 2, and the x axis.

**Answer =**16

**6-)**R is the region bounded by the curve $y = 1 − x^2$ and the x axis.

**Answer =**$\frac{4}{3}$

**7-)**R is the region bounded by the curve $y = −x^2 − 6x − 5$ and the x axis

**Answer =**$\frac{32}{3}$

**8-)**R is the region in the first quadrant bounded by the curve $y = 4 − x^2$ and the lines y = 3x and y = 0.

**Answer =**$\frac{19}{6}$

**9-)**R is the region bounded by the curve $y = \sqrt{x}$ and the lines y = 2 − x and y = 0.

**Answer =**$\frac{7}{6}$

**10-)**R is the region in the first quadrant that lies under the curve $y = \frac{16}{x}$ and that is bounded by this curve and the lines y = x, y = 0, and x = 8.

**Answer =**8(1+ln4)

**11-)**R is the region bounded by the curve $y = x^2−2x$ and the x axis. (Hint: Reflect the region across the x axis and integrate the corresponding function.)

**Answer =**$\frac{4}{3}$

**12-)**R is the region bounded by the curves $y = x^2 + 3$ and $y = 1 − x^2$ between x = −2 and x = 1.

**Answer =**12

**13-)**R is the region bounded by the curve $y = e^x$ and the lines y = 1 and x = 1.

**Answer =**e-2

**14-)**R is the region bounded by the curve $y = x^2$ and the line y = x.

**Answer =**$\frac{1}{6}$

**15-)**R is the region bounded by the curve $y = x^2$ and the line y = 4.

**Answer =**$\frac{32}{3}$

**16-)**R is the region bounded by the curves $y = x^3 − 6x^2$ and $y = −x^2$.

**Answer =**$\frac{625}{12}$

**17-)**R is the region bounded by the line y = x and the curve $y = x^3$.

**Answer =**$\frac{1}{2}$

**18-)**R is the region in the first quadrant bounded by the curve $y = x^2 + 2$ and the lines y = 11 − 8x and y = 11.

**Answer =**$\frac{40}{3}$

**19-)**R is the region bounded by the curves $y = x^2 − 3x + 1$ and $y = −x^2 + 2x + 2$.

**Answer =**$\frac{11}{8}\sqrt{33}$

**20-)**R is the region bounded by the curves $y = x^3 − x$ and $y = −x^2 + x$.

**Answer =**$\frac{37}{12}$

## What is integral of x^3/4 | $\int x^{\frac{3}{4}}.dx=?$

## Tuesday, September 13, 2022

## $x.e^{-x}$ integral | What is integral of x.e^-x? | $\int x.e^{-x}.dx=?$

## $x.lnx^2$ integral | What is integral of x.lnx^2? | $\int x.lnx^2.dx=?$

You can reach integral of x.lnx^2 answer on this page.

**What is integral of ∫x.lnx^2.dx?**

Solution. In this case, the factor $X$ is easy to integrate, while the factor ln $x^2$ is simplified by differentiation. This suggests that you try integration by parts with:

**$\int x.lnx^2.dx=\frac{1}{2}.x^2(lnx^2-1)+C$**

## Monday, September 12, 2022

## What is integral of 1/x^2? What is the integral of one over x squared?

## Integral of 3x^2-√5x+2 | What is integral of ∫(3x^2-√5x+2).dx?

## What is integral of root x? ∫√x.dx=? Root x integrate

## Sunday, September 4, 2022

## Graph of x^lnx | What is x to lnx graph?

Good day to all dear friends. In this article, we will share the graph of x to the lnx. We wish you a good work ahead...

## What is the graph of x to the lnx?

**The graph plot of x to the lnx is as follows:**