Linked Questions

So I came across with this integral today in my midterm: $$ \int \frac {\tan(\pi x)\sec^2(\pi x)}2 $$ And I got two correct answers: $$\frac {\sec^2(\pi x)}{4\pi} +C$$ And $$\frac {\tan^2(\pi x)}{...

While solving problems on indefinite integrals many a times I get answers which are different from those given in my text book's answer keys page. I then verify my solution steps to ensure that even ...

$$\int x(x^2+2)^4\,dx $$ When we do this integration with u substitution we get $$\frac{(x^2+2)^5}{10}$$ as $u=x^2+2$ $du=2x\,dx$ $$\therefore \int (u+2)^4\,du = \frac{(x^2+2)^5}{10} + C$$ ...

There seems to be no way to get these expressions from both the methods to match. I know the arbitrary constant could vary in different methods but even the strcture of both these expressions are not ...

For example, determine $\int \left(\frac{1}{2x+1}\right)dx$. Given that $f(x)$ = $\ln(2x+1)$ and $f'(x)$ = $\ln\left(\frac{2}{2x+1}\right)$. Would this be $\frac{1}{2} \int\left (\frac{2}{2x+1}\...

I'm going through Stewart's Calculus, and in section 7.3, example 7 asks for $\int \tan^3(x)dx$. In the book, the solution is $\frac{\tan^2(x)}{2} - \ln|\sec x| + C$. This was obtained in the ...

I tried Evaluating $\int \sqrt{\dfrac{5-x}{x-2}}dx$ using two different methods and got two different results. Getting two different answers when tried using two different methods:- M-$1$: $$\int \...

I want to find the integral $$\int\frac{x}{2x-2}dx$$ This is just a simple question from my textbook. But there seems to be two ways of solving it. If I simplify it to: $$\int1+\frac{2}{2x-2}dx$$ I ...

Evaluate the integral $$ \int\frac{a^x}{\sqrt{1-a^{2x}}}\ dx $$ In a recent question, the OP asked what the problem was with their method and why it did not match the reference book. I tried the ...

I am trying to integrate $\frac{1-x}{(x+1)^2}$, but I get to answers for two different methods: First, $\frac{1-x}{(x+1)^2} = \frac{1-x+1-1}{(x+1)^2} = \frac{2}{(x+1)^2} - \frac{x+1}{(x+1)^2} = \frac{...

I tried solving the following integral using integral by parts : $$\int \frac{x^3}{(x^2+1)^2}dx$$ but I got a different answer from Wolfram Calculator This is the answer that I got : $$\int \frac{x^3}{...

In looking at the equality $$\int \frac{a}{b(c-x)}dx = \int dt$$ I obtained different answers via different methods. Via one method, I got $$- \frac{a}{b} \ln(c-x) = t+C$$ Via another, I got $$- \frac ...

I was solving the integral $$\int\frac{\sin x}{\cos^3x}dx$$ I used the substitution $$u=\cos x \qquad du=-\sin x\,dx \qquad -du=\sin x dx$$ So the integral has the form $$\int-\frac{du}{u^3}=\frac{1}{...

I am currently taking a calculus class. My teacher while teaching a specific type of indefinite integral told one mysteriously beautiful of the solving the integral. The general form of the integral ...

If you compute the integral: $$ \int \frac{2x}{(x+1)^2}.$$ by substitution (using $u=x+1$, then you will get. $$ 2 \ln | 1 + x | + \frac{2}{1+x} + C.$$ But if, instead, you use integration by parts: $...