arcsin(x) integral | What is integrate of arcsinx or sin^-1x?

Greetings dear friends. In this article, we will share with you what is the integrate of arcsin(x).

arcsin integrate

Integral of arcsin(x) = $x.arcsin(x)+\sqrt{1-x^2}+C$

>>> $arcsin(x)=sin^{-1}x$

>>> $\int arcsin(x).dx=\int sin^{-1}x.dx$

>>> $\int arcsin(x).dx=x.arcsin(x)+\sqrt{1-x^2}+C$

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

Hello everyone. This lesson we will tell the integrate of $\int \frac{x}{x-1}dx$.

What is the integral of x/x-1?

integral of x/x-1

First we separate the expression.

So;

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

So our answer;

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

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$\sqrt{1+\sqrt{1+\sqrt{x}}}$ 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.

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Product Rule for Derivative | How to take the multiplication 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:

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

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

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

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 $(\frac{x^3}{e^x})'$

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

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

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

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

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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.

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

You can reach integrate of e^1-x solution and answer.

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

integrate of e^1-x

Answer:

$\int e^{1-x}dx=?$

$\int e^{1-x}dx=-e^{1-x}+C$

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

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

What is the graph of tanx?

The plot of tan(x) is as follows:

tangent graph

graph of tan(x)

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

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$\int \frac{x^2+1}{x^2-1}dx=?$ | 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 : 

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

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

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

What is integral of ∫1/x√x.dx?

integral of 1/x√x

Question:

$\int \frac{1}{x\sqrt{x}}dx$

Solution:

$\int \frac{1}{x\sqrt{x}}dx=\int x^{-\frac{3}{2}}dx=-\frac{2}{\sqrt{x}}+C$

Answer:

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

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$\int sin^5{x}dx=?$ | 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: 

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

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

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

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

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

You can reach integral of $x.e^{x^2}$ solutions.

What is integral of $x.e^{x^2}$ ?

x.e^x2 integrate solution

Solution:

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

Substituting $u = x^2$ and $\frac{1}{2} du = x.dx$, you get

$\int x.e^{x^2}dx=\frac{1}{2}\int e^u = \frac{1}{2}e^u+C=\frac{1}{2}e^{x^2}+C$

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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.

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

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

What is the graph of cos?

The plot of cos(x) is as follows:

cos graph

graph of cosx

Both graphs are graphs of cos. You can use both. Thank you so much. We wish you success in your studies.

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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

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

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

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

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

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 = $\frac{37}{12}$

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

You can reach integral of x^3/4 answer on this page.

What is integral of ∫x^3/4.dx?

integral of x^3/4

$\int x^{\frac{3}{4}}.dx=\frac{4}{7}.x^{\frac{7}{4}}+C$

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

You can reach integral of x.e^-x answer on this page.

What is integral of ∫x.e^-x.dx?

x.e^-x integral

$\int x.e^{-x}.dx=-(x+1).e^{-x}+C$

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$x.lnx^2$ 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:

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

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

You can reach integral of 1/x^2 or x^-2 (one over x square) answer on this page.

What is integral of 1/x^2?

1/x^2 integral

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

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

You can reach integral of 3x^2-√5x+2 answer on this page.

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

integral of 3x^2-root 5x+2

$\int (3x^2-\sqrt{5x}+2).dx=x^3-\frac{2}{3}.x.\sqrt{5x}+2x$

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

You can reach integral of root x answer on the page.

What is integrate of root x?

root x integral

$\int \sqrt{x}.dx=\frac{2}{3}.x.\sqrt{x}$

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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:

graph of x^lnx

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Graph of x^e | What is graph of x to e?

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

What is the graph of x to the e?

The graph plot of x to the e is as follows:

x^e graph

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Graph of e^(-x) | The graph of e to the -x | e to minus x graph

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

What is the graph of e to the -x?

The graph plot of e to the -2 is as follows:

e to -x graph

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