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Recent questions in Applications of integrals

airesex2 2022-04-22

Integrate $\int \frac{\sin x \cos x}{\sin^{4} x + \cos^{4} x} d x$

Malachi Novak 2022-04-22

Integral of $\sin^{5} (x) \cos (x)$
I assumed I would do u-substitution where:
$u = \sin (x)$
$d u = \cos (x) d x$
Which would then cancel out the $\cos (x)$
And leave me with:
$\int u^{5} d u = \frac{u^{6}}{6} + C = \frac{\sin^{6} (x)}{6} + C$

gaitaprepeted05u 2022-04-21

Integrating $\int \frac{1}{\cos (θ)} d θ$

znacimavjo 2022-04-21

Integrate $\int_{0}^{π} \sqrt{\frac{1 + \cos (2 x)}{2}} d x$
It is obvious that by using trig identities we can come up with:
$\int_{0}^{π} \sqrt{\cos^{2} (x)} =$
$= \int_{0}^{π} | \cos (x) | d x$

Troszokd95 2022-04-21

Integrate $\int \frac{d x}{1 + \sin x}$
My attempt:
$= \int \frac{d x}{1 + \sin x}$
$= \int {\frac{d x}{1 + \frac{2 \tan (\frac{x}{2})}{1 + \tan^{2} (\frac{x}{2})}}}$
$= \int {\frac{1 + \tan^{2} (\frac{x}{2})}{{(1 + \tan (\frac{x}{2}))}^{2}}} d x$

Molecca89g 2022-04-21

How to compute the integral of $\int \sin (1 - x^{2}) d x$
I tried substitution, $u = 1 - x^{2}, d u = - 2 x d x, d x = - \frac{d u}{2 x}$ , yielding:
$\int \sin u (- \frac{d u}{2})$
$= - \frac{1}{2} \int \sin (u) d u$
$= - \frac{1}{2} \cos u$
$= - \frac{1}{2} \cos (1 - x^{2})$
WolframAlpha is showing something completely different, however. What's wrong with my solution?

Ari Sheppard 2022-04-19

Solve $\int_{| z | = 3} \tan (π z) d z$ using argument principle
$\int_{| z | = 3} \tan (π z) d z = \int \frac{d (\cos π z)}{(- π) \cos π z} d z = - 2 i (N - P)$
where N = num of zeroes inside C:|z| = 3 and P is num of poles inside C (Is this correct or should we also consider on C???)
zeroes for $\cos π z = - 0.5, - 1.5, - 2.5, 0.5, 1.5, 2.5 \Rightarrow N = 6$
No poles for $\cos π z$
value of given integral = $- 2 i (N - P) = - 2 i (6 - 0) = - 12 i$
Is this correct? pls correct me if i am doing wrong

roselineanne22 2022-04-18

Determine the coordinates of the centriod of the region bounded by the graphs f x =4 -x^2 and g(x) =x + 2

gabolzm6d 2022-04-14

Is $lim_{n \to \infty} \int_{0}^{\frac{π}{2}} \frac{x^{\frac{1}{n}} + \sin x}{\tan x + x^{\frac{1}{n}}} d x = \int_{0}^{\frac{π}{2}} \frac{1 + \sin x}{\tan x + 1} d x = 1.62 . .$ ?
Note: The LHS of this identity it's seems converges for every positive integer $n \geq 1$ over the range $(0, \frac{π}{2})$

Jefferson Pacheco 2022-04-13

Is there any (elementary) way to evaluate the integral
$\int_{0}^{\frac{π}{2}} {(\cos x)}^{a} {(1 - \cos x)}^{1 - a} d x$
for any $a \in (0, 1)$ ?

Elijah Schwartz 2022-04-13

How do I integrate the following:
$\int \frac{\sqrt{\tan a x}}{1 + \sqrt{\tan a x}} d x$

Ormezzani6cuu 2022-04-11

The problem is to prove that:
$\int_{- π}^{π} \int_{- π}^{π} (a \sin (x) + b \cos (x) + c \sin (y) + d \cos (y)) e^{- 2 \cos (y - x) + a \cos (x) - b \sin (x) + c \cos (y) - d \sin (y)} d x d y$
I have an idea to represent a function under the integral like the odd function, but I can't.
Also, I can prove, that: $\int_{- π}^{π} (a \cos (x) + b \sin (x)) e^{a \sin (x) - b \cos (x)} d x = 0$

pevljivuaosyc 2022-04-10

Find the integral: $\int - \sqrt{t^{2} + t - 5} d t$ .

vilitatelp014 2022-04-10

Find the integral: $\int \frac{1}{\sqrt{x^{2} + x + 1}} d x$ .

Kiersten Hodge 2022-04-10

Integrate $- \frac{e^{t}}{\sqrt{4 e^{2 t} - 1}}$ with respect to t.

lifretatox8n 2022-04-10

Integrate $\frac{2 \sqrt{\sin^{- 1} (6 t)}}{3 \sqrt{1 - 36 t^{2}}}$ with respect to t.

datomerki8a5yj 2022-04-10

Evaluate $\int (\frac{1}{x^{4}} - \frac{1}{2 x^{2}} - \frac{5 x}{2} - 3 \sin (2 x)) d x$ .

affizupdaftf3opg 2022-04-10

Evaluate $\int \frac{2 \cos (2 t)}{3 (\sin (2 t) - 2)^{9}} d t$ .

sembuang711q6 2022-04-10

Evaluate $\int \frac{5 \tan^{- 1} (4 x)^{10}}{3 (16 x^{2} + 1)} d x$ .

deformere692qr 2022-04-10

Integrate $\frac{8}{7} x^{3} \tan (x^{4} + 4) \sec (x^{4} + 4)$ with respect to x.

Applications of integrals are one of the popular but still complex fields that many students encounter not only in Math and Physics but in subjects like Programming and Engineering. For example, start with the applications of double integrals and you will see how it is useful in the construction of some building where the shape must be found or the gravity matters should be presented as the model. At the same time, if your questions are based on equations, consider geometric applications of definite integrals help that is also present below as you take a look through the relevant answers.

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