2
votes
4answers
124 views

Using integral definition to solve this integral

I'm trying to solve this question using the definition of integral: $$\int^5_2 (4-2x)dx$$ Definition of integral: We define first the inferior and superior sum: Let $f:[a,b]\to \mathbb R$ be a ...
1
vote
3answers
78 views

How to calculate $\int\sqrt{\frac{1-\sqrt x}{1+\sqrt x}} dx$?

How to calculate? $$\int\sqrt{\frac{1-\sqrt x}{1+\sqrt x}}\, \mathrm dx$$ I try to let $x=\cos^2 t$, then $$\sqrt{\frac{1-\sqrt x}{1+\sqrt x}}=\tan\frac t2,\; dx=-2\sin t\cos t\,\mathrm dt $$ so ...
0
votes
1answer
56 views

Compute $\int \frac{\sin x+\cos x}{\sqrt[3]{\sin x-\cos x}}\,\mathrm dx$

Compute $$\int \frac{\sin x+\cos x}{\sqrt[3]{\sin x-\cos x}}\,\mathrm dx$$ Let $t=\tan\frac x2$, then $$\sin x=\frac{2t}{1+t^2},\; \cos x=\frac{1-t^2}{1+t^2},\; \mathrm dx =\frac{2\mathrm ...
3
votes
3answers
111 views

How to solve $\int \frac{x^4 + 1 }{x^6 + 1}$?

How to solve $\int \frac{x^4 + 1 }{x^6 + 1}$ ? The numerator is a irreducible polynomial so I can't use partial fractions. I tried the substitutions $t = x^2, t=x^4$ and for the formula $\int ...
1
vote
4answers
102 views

Antiderivative of $\quad$$t^2e^{-\frac{1}{2}t^2}$

What is the antiderivative of $\quad$$t^2e^{-\frac{1}{2}t^2}$ ? $\displaystyle\int t^2e^{-\frac{1}{2}t^2}\,dt=\displaystyle\int{t}_{}te^{-\frac{1}{2}t^2}\,dt=-te^{-\frac{1}{2}t^2}\Big|_{?}^?+\int ...
0
votes
1answer
29 views

–°hange the order of integration in the double integral

I have to change the order of integration in this double integral I've decided to divide it in two similar areas D1 and D2 And I've got the following result Can You chech it and state my ...
0
votes
1answer
57 views

Cannot solve indefinite integral

can You help me with this indefinite integral $$\int \dfrac{2^{x + 1} + 5^x}{10^x} dx$$ What to use. direct integration or substitute of variables Thanks
0
votes
2answers
52 views

Help me with indefinite integral

Can You help me with it $$\int\ \frac{tg^2 x}{ \cos^2x}\ dx$$ I've already tried to decompose tg. But I don't know what to do after that.
5
votes
1answer
4k views

Indefinite Integral for $\cos x/(1+x^2)$

I have been working on the indefinite integral of $\cos x/(1+x^2)$. $$ \int\frac{\cos x}{1+x^2}\;dx\text{ or } \int\frac{\sin x}{1+x^2}\;dx $$ are they unsolvable(impossible to solve) or is there a ...