Solve Integral Calculus Problems with Ease

College
Calculus and Analysis
Integral Calculus

How do you name the curve given by the conic $r = \frac{4}{1 + \cos θ}$ ?

rabunztr 2022-02-09

How do I find the directrix and focus of a parabola?

Gerald Ritter 2022-01-30

Solve this trigonometric equation, $\tan x + \tan 2 x = 1 - \tan x \tan 2 x$

Turnseeuw 2022-01-30

Integrate $\int_{0}^{1} \sin^{- 1} {\frac{x^{2}}{1 + x^{2}}} d x$

Solve This plox my Exam is going on soo

minikim38 2022-01-29

How to calculate the indefinite integral of $\frac{1}{x^{\frac{2}{3}} (1 + x^{\frac{2}{3}})}$ ?
I substituted,
$t = \frac{1}{x^{\frac{1}{3}}}$
$\frac{d t}{d x} = - \frac{1}{3 x^{\frac{4}{3}}}$
$\frac{d t}{d x} = - \frac{t^{4}}{3}$
Rewriting the question,
$\int \frac{d x}{x^{\frac{2}{3}} + x^{\frac{4}{3}}}$
$- \frac{1}{3} \int \frac{d t}{t^{4} (\frac{1}{t^{2}} + \frac{1}{t^{4}})}$
We have,
$- \frac{1}{3} \int \frac{d t}{t^{2} + 1}$
$- \frac{1}{3} \tan^{- 1} t + C$
$- \frac{1}{3} \tan^{- 1} (\frac{1}{x^{\frac{1}{3}}}) + C$
But the answer given is
$3 \tan^{- 1} x^{\frac{1}{3}} + C$
Where am I wrong?

Natasha Henry 2022-01-28

The usual arctangent addition formula works only when the arguments are in the proper range, namely:
$\arctan x + \arctan y = \arctan \frac{x + y}{1 - x y}, | x y | < 1$
However for the other arguments we need to add some integer multiple of $π$ .

Mlejd5 2022-01-28

Consider the function
$f (x) = \frac{\sin (x)}{\sin ((2 k + 1) x)}$
with k a positive integer. It seems that f is strictly increasing in $[0, \frac{π}{2 (2 k + 1)}]$ .Is there some easy proof of this monotonicity property that does not invole differentiation (through suitable inequalities)?

amevaa0y 2022-01-27

How can I evaluate the following integral?
$\int \frac{\cos {x}}{1 + \sin {2 x}} d x$
I tried the following way, but I was not able to proceed further:
$I = \int \frac{\cos {x}}{{(\sin {x} + \cos {x})}^{2}} d x$
$= \int \frac{\sec {x}}{{(1 + \tan {x})}^{2}} d x$

Connor Randall 2022-01-27

How do I integrate $\sin θ + \sin θ \tan^{2} θ$ ?

Natasha Gill 2022-01-26

I was trying to solve the integral $\int \sin (2 x) \sin (4 x) d x$ . Heres

kramtus51 2022-01-04

Evaluate the definite integral exactly
$\int_{0}^{1} {(e^{5 x} - 5 x)}^{3} (e^{5 x} - 1) d x$

Mary Buchanan 2021-12-25

What is $10 < 30 + 10 < 30$ ? Answer in polar form. Note that the angle is measured in degrees here and need to be in Phasors.

Pamela Meyer 2021-12-21

Solve and give correct answer for this taylor series of ln(1+x)?

Sapewa 2021-12-21

Find the charge flowing through a device if the current is:
a) $i (t) = 3 A, q (0) = 1 C$
b) $i (t) = (2 t + 5) m A,$
$q (0) = 0 (c) i (t) = 20$
$\cos (10 t + \frac{π}{6}) μ A, q (0) = 2 μ C (d) A, i (t) = 10 e^{- 30 t} \sin 40 t A, q (0) = 0$

expeditiupc 2021-12-19

For the integral state the number of the integration formula and the values of the constans a and b so that the formula fits the integral
$\int \frac{1}{x^{2} (4 x - 3)} d x$

Arthur Pratt 2021-12-19

Using polar coordinates, evaluate the integral
$\int \int_{R} \sin (x^{2} + y^{2}) d A$ where R is region
$4 \leq x^{2} + y^{2} \leq 64$

zeotropojd 2021-12-19

Find the curve's length over the specified range. $r = 8 + 8 \cos θ$ on the interval $0 \leq θ \leq π$

sputneyoh 2021-12-18

Locate the centroid x of the area.

jamessinatraaa 2021-12-17

What is the integral of $\sec^{3} (x)$ ?

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Integral Calculus

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