Evaluate each integral. (This is the same as finding the most general antiderivative of the function inside the integral sign.)
(a) \(\displaystyle\int x^3\,dx\)
(b) \(\displaystyle\int e^x\,dx\)
(c) \(\displaystyle\int \frac{1}{x}\,dx\)
(d) \(\displaystyle\int \sqrt{x}\,dx\)
(e) \(\displaystyle\int \sqrt[3]{x^2}\,dx\)
(f) \(\displaystyle\int dx\)
(g) \(\displaystyle\int 0\,dx\)
(h) \(\displaystyle\int 2\,dx\)
(i) \(\displaystyle\int \pi^2\,dx\)
(j) \(\displaystyle\int e^3\,dx\)
(k) \(\displaystyle\int x^{99}\,dx\)
(l) \(\displaystyle\int \frac{1}{x^4}\,dx\)
(m) \(\displaystyle\int 3^x\,dx\)
(n) \(\displaystyle\int 5\cdot 2^x\,dx\)
Find the most general antiderivative of the function.
(a)
\(f(x) = 1 - 3x^2 + e^x\)
(b)
\(\displaystyle f(x) = \sqrt[4]{x} - \frac{1}{x^2}\)
(c)
\(\displaystyle f(x) = \frac{1+x-x^2}{x}\)
Find the most general antiderivative of the function.
(a)
\(f(t) = 2t(2-t)^2\)
(b)
\(\displaystyle f(x) = \frac{x^2+2x-4}{\sqrt{x}}\)
(c)
\(f(x) = (2x-1)(x+3)\)
Find the most general antiderivative of the function.
(a)
\(f(\theta) = \pi - \sec\theta\tan\theta\)
(b)
\(\displaystyle f(x) = e^x + \cos x + \frac{1}{\cos^2 x}\)
(c)
\(\displaystyle f(x) = 14\sqrt[4]{x^3} + \frac{2}{1+x^2}\)
Initial Value Problems
Find \(f(2)\) given that the graph of \(f\) passes through the point \((1,6)\) and that the slope of its tangent line at \((x, f(x))\) is \(2x+1\).
6
Find f from Conditions
Find the function \(f\) in each case.
(a)
If \(f'(x) = 1-6x\) and \(f(1)=8\), find \(f\).
▶ Solution
(b)
If \(f''(x) = \sin x + \cos x\) and \(f(0)=3,\ f'(0)=4\), find \(f\).
▶ Solution
(c)
If \(f''(t) = 2e^t + 3\sin t\) and \(f(0)=0,\ f(\pi)=0\), find \(f\).
▶ Solution
A particle is moving with the following data. Find the position \(s(t)\) of the particle.
\[ v(t) = 3\sqrt{t}+1, \quad s(4) = 10 \]
8
Initial Value Problem
Solve the initial value problem.
(a)
\(f'(x) = 1+3x^2,\quad f(-1)=2\)
▶ Solution
(b)
\(\displaystyle f'(x) = 2x - \frac{1}{x},\quad f(1)=5\)
▶ Solution
(c)
\(\displaystyle f'(x) = \frac{1}{\sqrt{x}},\quad f(4)=2\)
▶ Solution
Bonus
Evaluate the integral.
\[ \int \left(\sin(3x) - e^{5x}\right)dx \]