Evaluate each of the following indefinite integrals. That is, write down a formula for the family of all antiderivatives of each integrand; don’t forget the \(+\!C.\) Recall that a proposed antiderivative can be verified by differentiating it, checking that the result is identical to the original integrand. Software capable of symbolic integration can also be used to verify an antiderivative.
Basic Integrals
\(\displaystyle \int 7x^{6} \,\mathrm{d}x\)
\(\displaystyle \int x^{6} \,\mathrm{d}x\)
\(\displaystyle \int \tfrac{1}{7}x^{6} \,\mathrm{d}x\)
\(\displaystyle \int x^4\!+\!x^3\!+\!x^2\!+\!x\!+\!1\!+\!\tfrac{1}{x^2}\!+\!\tfrac{1}{x^3}\!+\!\tfrac{1}{x^4} \,\mathrm{d}x\)
\(\displaystyle \int \sqrt{x} + 2\sqrt[3]{x} + \sqrt{5}{x^6} \,\mathrm{d}x\)
\(\displaystyle \int \frac{1}{\sqrt[3]{x}} \,\mathrm{d}x\)
\(\displaystyle \int 7x^{\pi} \,\mathrm{d}x\)
\(\displaystyle \int \sin(x) \,\mathrm{d}x\)
\(\displaystyle \int \cos(x) \,\mathrm{d}x\)
\(\displaystyle \int \tan(x)\sec(x) \,\mathrm{d}x\)
\(\displaystyle \int \sec^2(x) \,\mathrm{d}x\)
\(\displaystyle \int \sin(x)-\cos(x) \,\mathrm{d}x\)
\(\displaystyle \int \sec(x)\bigl(\tan(x)+\sec(x)\bigr) \,\mathrm{d}x\)
\(\displaystyle \int \bigl(7-x^3\bigr)^2 \,\mathrm{d}x\)
\(\displaystyle \int \bigl(x-5\bigr)\bigl(1-2x\bigr) \,\mathrm{d}x\)
\(\displaystyle \int \sqrt[3]{x}\bigl(x^2-11\bigr) \,\mathrm{d}x\)
\(\displaystyle \int \sqrt{x^2-8x+16} \,\mathrm{d}x\)
\(\displaystyle \int \frac{x^3-x}{\sqrt{x}} \,\mathrm{d}x\)
\(\displaystyle \int \frac{1+\frac{1}{x}}{x^2} \,\mathrm{d}x\)
Integrals that Suggest a Substitution
\(\displaystyle \int 7\sin\bigl(7x\bigr) \,\mathrm{d}x\)
\(\displaystyle \int 3x^2\sin\bigl(x^3\bigr) \,\mathrm{d}x\)
\(\displaystyle \int x^4\sin\bigl(x^5\bigr) \,\mathrm{d}x\)
\(\displaystyle \int \bigl(4x^3\!-\!2\bigr)\sin\bigl(x^4\!-\!2x\bigr) \,\mathrm{d}x\)
\(\displaystyle \int \bigl(4x^3\!-\!2\bigr)\sqrt{x^4\!-\!2x} \,\mathrm{d}x\)
\(\displaystyle \int \frac{4x^3\!-\!2}{\bigl(x^4\!-\!2x\bigr)^2} \,\mathrm{d}x\)
\(\displaystyle \int \frac{4x^3\!-\!2}{\sqrt{x^4\!-\!2x}} \,\mathrm{d}x\)
\(\displaystyle \int \frac{4x^3\!-\!2}{\sin^2\bigl(x^4\!-\!2x\bigr)} \,\mathrm{d}x\)
\(\displaystyle \int \sec^2(x)\sin\bigl(\tan(x)\bigr) \,\mathrm{d}x\)
Integral Potpourri
\(\displaystyle \int t^2\frac{\sqrt{t^3\!-\!1}}{t^3\!-\!1} \,\mathrm{d}t\)
\(\displaystyle \int \sec\bigl(5\theta\bigr)\tan\bigl(5\theta\bigr) \,\mathrm{d}\theta \)
\(\displaystyle \int x^{100}\!-\!\frac{1}{x^{100}} \,\mathrm{d}x \)
\(\displaystyle \int \sqrt[2.3]{x} \,\mathrm{d}x \)
\(\displaystyle \int \frac{x^3\,\mathrm{d}x}{\bigl(4-x^2\bigr)^3} \)
\(\displaystyle \int \frac{\sin(t)\,\mathrm{d}t}{1\!-\!\sin^2(t)} \)
\(\displaystyle \int \frac{\sin(2\theta)\,\mathrm{d}\theta}{\sin(\theta)} \)
\(\displaystyle \int \bigl(x\!-\!2\bigr)^9 \,\mathrm{d}x \)
\(\displaystyle \int \bigl(x-1\bigr)^5\bigl(x\!-\!3\bigr)^7 \,\mathrm{d}x \)
\(\displaystyle \int x^2\!-\!7^2 \,\mathrm{d}x \)
\(\displaystyle \int \bigl(7t-2\bigr)^{4} \,\mathrm{d}t \)
\(\displaystyle \int x\bigl(6x^2-8\bigr)^{7} \,\mathrm{d}x \)
\(\displaystyle \int \frac{t^{6}\!-\!1}{(t^{7}\!-\!7t\!+\!2)^5} \,\mathrm{d}t \)
\(\displaystyle \int x\sqrt{x^2\!-\!1} \,\mathrm{d}x \)
\(\displaystyle \int x^3\sqrt{x^2\!-\!1} \,\mathrm{d}x \)
\(\displaystyle \int x^7\sqrt{x^2\!-\!1} \,\mathrm{d}x \)
\(\displaystyle \int \frac{1}{x^2\!+\!10x\!+\!25} \,\mathrm{d}x \)
\(\displaystyle \int \frac{(y\!-\!1)\,\mathrm{d}y }{\sqrt{y^2\!-\!2y\!-\!1}} \)
\(\displaystyle \int \frac{\sqrt{1+\sqrt{t}}}{\sqrt{t}} \,\mathrm{d}t \)
\(\displaystyle\int \tfrac{1}{2} \sec^2\bigl(3x-7\bigr) \,\mathrm{d}x\)
\(\displaystyle\int \frac{\mu-\mu^2}{\mu-1} \,\mathrm{d}\mu \)
\(\displaystyle\int \frac{1}{\sqrt[5]{t^2}} \,\mathrm{d}t \)
\(\displaystyle \int \tan(y)\sqrt{\cos(y)} \,\mathrm{d}x \)
\(\displaystyle \int \sqrt[3]{2+5\omega} \,\mathrm{d}\omega \)
\(\displaystyle \int \cos^3\bigl(\theta\bigr) \,\mathrm{d}\theta \)
\(\displaystyle \int \sqrt{x^7+4x^5} \,\mathrm{d}x \)
\(\displaystyle \int \cot(y) \,\mathrm{d}y \)
\(\displaystyle \int \cos(\theta)\sqrt{1\!+\!\sin(\theta)} \,\mathrm{d}\theta \)
\(\displaystyle \int \tan(x)\tan\bigl(\sec(x)\bigr)\sec(x) \,\mathrm{d}x \)
\(\displaystyle \int \csc^2(\tfrac{\pi}{6}-\theta) \,\mathrm{d}\theta \)
\(\displaystyle \int \cos(\theta)\sin^9(\theta) \,\mathrm{d}\theta \)
\(\displaystyle \int \tan(\theta)\sec^5(\theta) \,\mathrm{d}\theta \)
\(\displaystyle \int \cos^3(t) \,\mathrm{d}t \)
\(\displaystyle \int \sec(4t)\cot(4t) \,\mathrm{d}t \)
\(\displaystyle \int \sin(x)\cos(x) \,\mathrm{d}x \)
\(\displaystyle \int x^6\sec^2(x^7) \,\mathrm{d}x \)
\(\displaystyle \int (4x\!-\!2)\sec\bigl(x^2\!-\!x\bigr)\tan\bigl(x^2\!-\!x\bigr) \,\mathrm{d}x \)
\(\displaystyle \int \sin(x)\sin\bigl(\cos(x)\bigr) \,\mathrm{d}x \)
\(\displaystyle\int \sqrt{3x-7} \,\mathrm{d}x\)
\(\displaystyle\int \frac{\mathrm{d}x}{(3x-7)^2} \)
\(\displaystyle\int \cos(7x) \,\mathrm{d}x\)
\(\displaystyle\int x\cos\!\big(7x^2\big) \,\mathrm{d}x\)
\(\displaystyle\int x^3\cos\!\bigl(7x^4\bigr) \,\mathrm{d}x\)
\(\displaystyle\int x^3\bigl(x^2+1\bigr)^9 \,\mathrm{d}x\)
\(\displaystyle\int \cos(x)\cos\!\big(\sin(x)\big) \,\mathrm{d}x\)
\(\displaystyle\int \sin(x)\cos^{9}(x) \,\mathrm{d}x\)
\(\displaystyle\int x\sqrt{3x-7} \,\mathrm{d}x\)
\(\displaystyle\int x\sec^2\bigl( 3x^2-9 \bigr) \,\mathrm{d}x\)
\(\displaystyle\int \frac{1+\cot^2(x)}{(1+\cot(x))^2} \,\mathrm{d}x\)
\(\displaystyle\int \frac{\sin\bigl(\frac{1}{x^2}\bigr)}{x^3} \,\mathrm{d}x\)
\(\displaystyle\int \sqrt{\tan(x)\sec^4(x)} \,\mathrm{d}x\)