I have been trying to study for a test on monday but I can't do any of the basic problems. I know what to do but I am just not good enough at math to get the proper answer.
I am supposed to use part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. I am given integral $\hskip1in \int_x^\pi\cos(\sqrt{t}) \; dt\tag{1}$ I know that if it is from $\pi$ to $x$ all I do is replace $t$ with $x$ and I get the answer. There are no examples in this book of how to do this so I do not know how to do it.
Next I am supposed to evaluate the integrals and I got to a point in the problems where I can't do any of them after number 27 out of 45.
$\int_0^\pi (5e^x + 3\sin(x))\tag{2}$ I just can't think of how to get the integral for this.
$\int_1^4\frac{4+6u}{\sqrt{u}}\tag{3}$ I don't know substitution yet so I don't know how to do this one either. I know I could try and subtract exponent u's but the 4 has no u so it is not possible I think.
$\int_0^1 x(\sqrt[3]{x} + \sqrt[4]{x}) \; dx\tag{4}$ Again no idea what to do here, I made it into $x^{3/2} + x^{5/4}$ but that is still wrong after I integrate that.
$\int_1^2\left(\frac{x}{2} - \frac{2}{x}\right)\;dx\tag{5}$ I do not know what to do with this one either I tried to make it $\frac{1}{2x}$ and something else I think it might integrate to $\frac{1}{3}x^{3/2} - 2\ln(x)$ but I can't get a proper answer out of that.
$\int_0^1 (x^{10} + 10^x)$ I tried many things on this but was never right. $\frac{1}{11} x^{11} + 10x^{x+1}\tag{6}$ is not right but should be to me.
$\int_0^{\pi/4} \frac{1+\cos^2(x)}{\cos^2(x)}\tag{7}$ I tried $\frac{1}{\cos^2(x)}$ but to me that means nothing and I am left with maybe $\ln(\cos^2(x)) + 1$ but that is wrong.