The integral of a constant B with respect to x is Bx+A, where A is constant. Applying this rule to the constant function y(x)=0, \(\displaystyle{\int_{{{a}}}^{{{b}}}}{0}{\left.{d}{x}\right.}={0}+{A}={A}\).

asked 2021-08-17

Use the Table of Integrals to evaluate the integral. (Use C for the constant of integration.)

\(\displaystyle\int{37}{e}^{{{74}{x}}}{\arctan{{\left({e}^{{{37}{x}}}\right)}}}{\left.{d}{x}\right.}\)

Inverse Trigonometric Forms (92): \(\displaystyle\int{u}{{\tan}^{{-{1}}}{u}}\ {d}{u}={\frac{{{u}^{{{2}}}+{1}}}{{{2}}}}{{\tan}^{{-{1}}}{u}}-{\frac{{{u}}}{{{2}}}}+{C}\)

\(\displaystyle\int{37}{e}^{{{74}{x}}}{\arctan{{\left({e}^{{{37}{x}}}\right)}}}{\left.{d}{x}\right.}\)

Inverse Trigonometric Forms (92): \(\displaystyle\int{u}{{\tan}^{{-{1}}}{u}}\ {d}{u}={\frac{{{u}^{{{2}}}+{1}}}{{{2}}}}{{\tan}^{{-{1}}}{u}}-{\frac{{{u}}}{{{2}}}}+{C}\)

asked 2021-08-15

Use the Table of Integrals to evaluate the integral. (Use C for the constant of integration.)

\(\displaystyle{7}{{\sin}^{{{8}}}{\left({x}\right)}}{\cos{{\left({x}\right)}}}{\ln{{\left({\sin{{\left({x}\right)}}}\right)}}}{\left.{d}{x}\right.}\)

no. 101. \(\displaystyle\int{u}^{{{n}}}{\ln{{u}}}{d}{u}={\frac{{{u}^{{{n}+{1}}}{\left\lbrace{\left({n}+{1}\right)}^{{{2}}}\right\rbrace}{\left[{\left({n}+{1}\right)}{\ln{{u}}}-{1}\right]}+{C}}}{}}\)

\(\displaystyle{7}{{\sin}^{{{8}}}{\left({x}\right)}}{\cos{{\left({x}\right)}}}{\ln{{\left({\sin{{\left({x}\right)}}}\right)}}}{\left.{d}{x}\right.}\)

no. 101. \(\displaystyle\int{u}^{{{n}}}{\ln{{u}}}{d}{u}={\frac{{{u}^{{{n}+{1}}}{\left\lbrace{\left({n}+{1}\right)}^{{{2}}}\right\rbrace}{\left[{\left({n}+{1}\right)}{\ln{{u}}}-{1}\right]}+{C}}}{}}\)

asked 2021-05-14

Use the Table of Integrals to evaluate the integral. (Use C for the constant of integration.)

\(\int37e^{74x}\arctan(e^{37x})dx\)

Inverse Trigonometric Forms (92): \(\int u\tan^{-1}u\ du=\frac{u^{2}+1}{2}\tan^{-1}u-\frac{u}{2}+C\)

\(\int37e^{74x}\arctan(e^{37x})dx\)

Inverse Trigonometric Forms (92): \(\int u\tan^{-1}u\ du=\frac{u^{2}+1}{2}\tan^{-1}u-\frac{u}{2}+C\)

asked 2021-05-27

Evaluate the indefinite integral as a power series.

\(\int \frac{\tan^{-1}x}{x}dx\)

\(f(x)=C+\sum_{n=0}^\infty\left( \dots \right)\)

What is the radius of convergence R?

\(\int \frac{\tan^{-1}x}{x}dx\)

\(f(x)=C+\sum_{n=0}^\infty\left( \dots \right)\)

What is the radius of convergence R?

asked 2021-11-15

Evaluate the integral. (Use C for the constant of integration.)

\(\displaystyle\int{e}^{{{4}\theta}}{\sin{{\left({5}\theta\right)}}}{d}\theta\)

\(\displaystyle\int{e}^{{{4}\theta}}{\sin{{\left({5}\theta\right)}}}{d}\theta\)

asked 2021-09-13

Evaluate the integral. (Use C for the constant of integration.)

\(\displaystyle\int{\ln{{\left(\sqrt{{x}}\right)}}}{\left.{d}{x}\right.}\)

\(\displaystyle\int{\ln{{\left(\sqrt{{x}}\right)}}}{\left.{d}{x}\right.}\)

asked 2021-05-02

Evaluate the integral. (Use C for the constant of integration.)

\(\int \ln (\sqrt x)dx\)

\(\int \ln (\sqrt x)dx\)