Integral Problems & Solutions - Explore with Us!

Recent questions in Applications of integrals

Spencer Lutz 2022-05-11

Compute $\int (\frac{x^{3}}{2} - \frac{2}{x^{3}} + \frac{3 \sin (x)}{2} + 2 \cos (3 x)) d x$ .

Amappyaccon22j7e 2022-05-11

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

bedblogi38am 2022-05-11

Compute $\int \frac{2}{3 t (\ln (4 t) - 1)} d t$ .

Defensorentx9 2022-05-11

Integrate $\frac{7}{2} \tan (2 x) \sec (2 x) \ln (\sec (2 x))$ with respect to x.

Daphne Fry 2022-05-11

Evaluate $\int \frac{5}{2} \cos (4 x) \ln (\sin (4 x) + 1) d x$ .

Jayden Mckay 2022-05-11

Integral $\frac{7}{6} \cos (5 t) \ln (\sin (5 t) + 2)$ with respect to t.

Amappyaccon22j7e 2022-05-11

Evaluate $\int \frac{7 x^{3}}{4 ((5 x^{4} + 4)^{2} + 1)} d x$ .

Brody Collins 2022-05-11

Integrate $\frac{8}{3} \sin (t) \sec^{3 / 2} (t)$ with respect to t.

hi hi2022-05-07

Determine a region whose area is equal to the given limit. Do not evaluate the limit. limit as n approaches infinity of $\sum_{i = 1}^{n} (2 / n) ((5 + (2 i) / n)^{10})$

Aaron Coleman 2022-05-03

Show that the integral is divergent $\int_{0}^{\infty} \frac{d x}{1 + x^{2} \sin^{2} x}$

predorsomwy 2022-05-02

Proving the double differential of z = -z implies $z = \sin x$
$\frac{d^{2} z}{{d x}^{2}} = - z$
implies z is of the form $a \sin x + b \cos x$ . Is there a proof for the same. I was trying to arrive at the desired function but couldn't understand how to get these trigonometric functions in the equations by integration. Does it require the use of taylor polynomial expansion of $\sin x or \cos x$ ?

deepa Varakala2022-05-01

Find the Maclaurin series for the function f (x)  (x2  2  x)5 and its radius of convergence.

esbagoar7kh 2022-04-29

I am trying to evaluate the integral
$\int_{0}^{\frac{π}{2}} \frac{\log (e^{i x} + e^{- i x})}{-} \log {2}} {e^{2 i x} - 1} e^{i x} d x$
I am wondering if a complex variable substitution the likes of $i u = e^{i x}$ , $d u = e^{i x} d x$ is justifiable? In my mind. when x=0,u=0 and when $x = \frac{π}{2}, u = 1$ which would make the integral
$- \int_{0}^{1} \frac{\log (i u - \frac{i}{u}) - \log (2)}{1 + u^{2}}, d u$

Rylan Sullivan 2022-04-25

Integrating $\int_{0}^{\frac{π}{4}} \frac{\sin x + \cos x}{\sin^{4} x + \cos^{2} x} d x$

Noe Wade 2022-04-24

Integrate $\int \frac{d x}{\sqrt{\sin^{3} x \sin (x + α)}}$
$\int \frac{1}{\sqrt{\sin^{3} x \sin (x + α)}} d x = \int \frac{1}{\sin x \sqrt{\sin x \sin (x + α)}} d x$
$= \frac{1}{\sin α} \int \frac{\sin (x + α - x)}{\sin x \sqrt{\sin x \sin (x + α)}} d x = \frac{1}{\sin α} \int \frac{\sin (x + α) \cos x - \cos (x + α) \sin {x}}{\sin x \sqrt{\sin x \sin (x + α)}} d x$
$= \frac{1}{\sin α} \int \frac{\sin (x + α) \cos x}{\sin x \sqrt{\sin x \sin (x + α)}} d x - \frac{1}{\sin α} \int \frac{\cos (x + α) \sin {x}}{\sin x \sqrt{\sin x \sin (x + α)}} d x$
Is it possible to proceed further and complete the integration or what is the right substitution to find the solution ?

dreangannaa 2022-04-24

Integrate $\int x \cos (x) \sin (x) d x$
I've been trying to integrate by parts, but I can't! I know there's a trigonometric function about $\cos (2 x)$ but I don't know how to integrate that function wit $\sin x$

zamenjenot7k 2022-04-23

integral $\int \frac{1 - a \cos x}{1 + a^{2} - 2 a \cos x} d x$
My trouble: Using software, i found that if $| a | > 1$ , the integral is equal to
$\arctan (\frac{x}{2}) + \frac{1 - a^{2}}{| a - 1 | | a + 1 |} \arctan (\frac{| a + 1 |}{| a - 1 |} \tan (\frac{x}{2})) + C$
If $| a | < 1$ , then the antiderivate is
$x - \arctan (\frac{x}{2}) + \frac{1 - a^{2}}{| a - 1 | | a + 1 |} \arctan (\frac{| a + 1 |}{| a - 1 |} \tan (\frac{x}{2})) + C$

Travis Caldwell 2022-04-22

Integral n for $\sin^{n} (x) + \cos^{n} (x) = 1 - \frac{n}{2} \sin^{2} (x) \cos^{2} (x)$ is n=4,6.
I have proved for n=4,6 it is true, for other integer, I tried to check range of functions on left and right side. This is not as effective, and I failed. Please suggest a method in this.

haguemarineo6h 2022-04-22

Integral of $\int {\frac{\cos^{4} x + \sin^{4} x}{\sqrt{1 + \cos 4 x}} d x}$

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.

Integral Calculus

Master Integral Application Problems