Help to Integral Calculus Solutions & Examples

Evaluate the following integral.
${\displaystyle \int x^{-3}(x+1)dx}$

Give the correct answer and solve the given equation $V=T(m,n),$
The set of lineartrans formations
$T:R^m\to R^n,$
together with the usualadditionand scalar multiplication of functions.

The given initial-value problem.
$y^{″}-2y^{\prime }-8y=5$
$y(0)=1$
$y^{\prime }(0)=0$

Evaluate the surface integral
${\displaystyle {\int }_{S}F\cdot dS}$
for the given vector field F and the oriented surface S. In other words, find the flux of F across S. For closed surfaces, use the positive (outward) orientation.
F(x,y,z)=xi+2yj+3zk
S is the cube with vertices ${\displaystyle (\pm 1,\pm 1,\pm 1)}$

Curves to parametric equations Find parametric equations for the following curves. Include an interval for the parameter values. Answers are not unique. The line segment starting at $P(-1,-3)\text{and ending at}Q(6,-16)$

Find the linear or affine transformations that satisty the desired properties and write it in the form $T(x)=Ax+b:$
The transformation $T:{\mathbb{R}}^2\to {\mathbb{R}}^2$ sending the origin to itself and a triangle of vertices
$(0,0),(1,0),(0,1)$ to a triangle of verices (0, 0),
$(\sqrt{\frac{2}{2}},\sqrt{\frac{2}{2}}),(-\sqrt{\frac{2}{2}},\sqrt{2}).$

Give the correct answer and solve the given equation ${\displaystyle \frac{dy}{dx}=\frac{y^2-1}{x^2-1},y\left(2\right)=2}$

What is a parametric curve? $x(t)=t3$
$y(t)=t$

Find the following integral.
${\displaystyle \int \frac{x^2-1}{x^2}dx}$

Give the correct answer and solve the given equation
Let ${\displaystyle p\left(x\right)=2+x\text{and}q\left(x\right)=x}$. Using the inner product $⟨p,q⟩={\int }_{-1}^{1}pqdx$ find all polynomials ${\displaystyle r\left(x\right)=a+bx\in P1\left(R\right)P}$
(R) such that {p(x), q(x), r(x)} is an orthogonal set.

Verify that $\Vert r^t(s)\Vert =1$.Consider the helix represented investigation by the vector-valued function $r(t)=<2\cos t,2\sin t,t>$.

(a) ${\displaystyle \frac{dy}{dx}-1+y^2}$

True or False: In many applications of definite integrals, the integral is used to compute the total amount of a varying quantity.

You measure the weight of 49 dogs , and find they have a mean weight of 72 ounces. Let's say the population standard deviation is 13.1 ounces. Based on this, construct a ${\displaystyle 90\mathrm{\%}}$ confidence interval for the true population mean dog weight.
Give your answers as decimals, to two places

The function ${\displaystyle \left(9h\right)=8e^{-0.4h}}$ can be used to determine the milligrams D of a certain drug in a patient's bloodstream h hours after the drug has been given. How many milligrams (to two decimals) will be resent in 7 years?

Consider the helix represented investigation by the vector-valued function $r(t)=<2\cos t,2\sin t,t>$. Solve for t in the relationship derived in part (a), and substitute the result into the original set of parametric equations. This yields a parametrization of the curve in terms of the arc length parameter s.

Solve the equation graphically in the given interval. State each answer rounded to two decimals.
${\displaystyle x^{\frac{1}{3}}-x=0,[-3,3]}$
$x=?$

Rational exponents evaluate each expression.
(a) ${\displaystyle {27}^{1\text{/}3}}$
(b) ${\displaystyle {(-8)}^{1\text{/}3}}$
(c) ${\displaystyle -{\left(\frac{1}{8}\right)}^{1\text{/}3}}$

If a and are any odd integers, then $a^2+b^2$ is even.
Prove?

That parametric equations contain more information than just the shape of the curve. Write a short paragraph explaining this statement. Use the following example and your answers to parts (a) and (b) below in your explanation. The position of a particle is given by the parametric equations $x=\sin (t)andy=\cos (t)$ where t represents time. We know that the shape of the path of the particle is a circle. a) How long does it take the particle to go once around the circle? Find parametric equations if the particle moves twice as fast around the circle. b) Does the particle travel clockwise or counterclockwise around the circle? Find parametric equations if the particle moves in the opposite direction around the circle.