Separation of Variables

Author: John J Weber III, PhD Corresponding Textbook Sections:

• Section 2.2 – Separation of Variables

Expected Educational Results

• Objective 5–1: I understand when the method of separation of variables is appropriate to solve ordinary differential equations.

• Objective 5–2: I can solve ordinary differential equations using the method of separation of variables.

• Objective 5–3: I can solve initial value problems using the method of separation of variables.

Calculus I

Derivatives

Differentiation Rules

General Formulas

• $\frac{d}{dx}c=0$$\displaystyle \frac{d}{dx}c=0$, where $c$$c$ is any constant.

• $\frac{d}{dx}\left[cf\left(x\right)\right]=c{f}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(x\right)$$\displaystyle \frac{d}{dx}[cf(x)]=cf^{\,\prime}(x)$, where $c$$c$ is any constant.

• $\frac{d}{dx}\left[f\left(x\right)±g\left(x\right)\right]={f}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(x\right)±{g}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(x\right)$$\displaystyle \frac{d}{dx}[f(x)\pm g(x)]=f^{\,\prime}(x)\pm g^{\,\prime}(x)$

• $\frac{d}{dx}\left[f\left(x\right)g\left(x\right)\right]={f}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(x\right)g\left(x\right)+{g}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(x\right)f\left(x\right)$$\displaystyle \frac{d}{dx}[f(x)g(x)]=f^{\,\prime}(x)g(x)+g^{\,\prime}(x)f(x)$

• $\frac{d}{dx}\frac{f\left(x\right)}{g\left(x\right)}=\frac{{f}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(x\right)g\left(x\right)-{g}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(x\right)f\left(x\right)}{\left[g\left(x\right){\right]}^{2}}$$\displaystyle \frac{d}{dx}\frac{f(x)}{g(x)}=\frac{f^{\,\prime}(x)g(x)-g^{\,\prime}(x)f(x)}{[g(x)]^2}$

• $\frac{d}{dx}f\left(g\left(x\right)\right)={f}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(g\left(x\right)\right){g}^{\phantom{\rule{0.167em}{0ex}}\prime }\left(x\right)$$\displaystyle \frac{d}{dx}f(g(x))=f^{\,\prime}(g(x))g^{\,\prime}(x)$

• $\frac{d}{dx}{x}^{n}=n{x}^{n-1}$$\displaystyle \frac{d}{dx}x^n=nx^{n-1}$\index{differentiation rules!power rule}

Exponential and Logarithmic Formulas

• $\frac{d}{dx}{e}^{x}={e}^{x}$$\displaystyle \frac{d}{dx}e^x=e^x$

• $\frac{d}{dx}{a}^{x}=\mathrm{ln}\left(a\right){a}^{x}$$\displaystyle \frac{d}{dx}a^x=\ln{(a)}a^x$, $a>0$$a>0$, $a\ne 1$$a\ne 1$

• $\frac{d}{dx}\mathrm{ln}\left(x\right)=\frac{1}{x}$$\displaystyle \frac{d}{dx}\ln{(x)}=\frac{1}{x}$

• $\frac{d}{dx}{\mathrm{log}}_{b}\left(x\right)=\frac{1}{x\mathrm{ln}\left(b\right)}$$\displaystyle \frac{d}{dx}\log_b{(x)}=\frac{1}{x\ln{(b)}}$, $b>0$$b>0$, $b\ne 1$$b\ne 1$

Trigonometric Formulas

• $\frac{d}{dx}\mathrm{sin}\left(x\right)=\mathrm{cos}\left(x\right)$$\displaystyle \frac{d}{dx}\sin{(x)}=\cos{(x)}$

• $\frac{d}{dx}\mathrm{cos}\left(x\right)=-\mathrm{sin}\left(x\right)$$\displaystyle \frac{d}{dx}\cos{(x)}=-\sin{(x)}$

• $\frac{d}{dx}\mathrm{tan}\left(x\right)={\mathrm{sec}}^{2}\left(x\right)$$\displaystyle \frac{d}{dx}\tan{(x)}=\sec^2{(x)}$

• $\frac{d}{dx}\mathrm{cot}\left(x\right)=-{\mathrm{csc}}^{2}\left(x\right)$$\displaystyle \frac{d}{dx}\cot{(x)}=-\csc^2{(x)}$

• $\frac{d}{dx}\mathrm{sec}\left(x\right)=\mathrm{sec}\left(x\right)\mathrm{tan}\left(x\right)$$\displaystyle \frac{d}{dx}\sec{(x)}=\sec{(x)}\tan{(x)}$

• $\frac{d}{dx}\mathrm{csc}\left(x\right)=-\mathrm{csc}\left(x\right)\mathrm{cot}\left(x\right)$$\displaystyle \frac{d}{dx}\csc{(x)}=-\csc{(x)}\cot{(x)}$

Inverse Trigonometric Formulas

• $\frac{d}{dx}\mathrm{arcsin}\left(x\right)=\frac{1}{\sqrt{1-{x}^{2}}}$$\displaystyle \frac{d}{dx}\arcsin{(x)}=\frac{1}{\sqrt{1-x^2}}$

• $\frac{d}{dx}\mathrm{arccos}\left(x\right)=-\frac{1}{\sqrt{1-{x}^{2}}}$$\displaystyle \frac{d}{dx}\arccos{(x)}=-\frac{1}{\sqrt{1-x^2}}$

• $\frac{d}{dx}\mathrm{arctan}\left(x\right)=\frac{1}{1+{x}^{2}}$$\displaystyle \frac{d}{dx}\arctan{(x)}=\frac{1}{1+x^2}$

• $\frac{d}{dx}\text{arccot}\left(x\right)=-\frac{1}{1+{x}^{2}}$$\displaystyle \frac{d}{dx}\text{arccot}{(x)}=-\frac{1}{1+x^2}$

• $\frac{d}{dx}\text{arcsec}\left(x\right)=\frac{1}{|x|\sqrt{{x}^{2}-1}}$$\displaystyle \frac{d}{dx}\text{arcsec}{(x)}=\frac{1}{|x|\sqrt{x^2-1}}$

• $\frac{d}{dx}\text{arccsc}\left(x\right)=-\frac{1}{|x|\sqrt{{x}^{2}-1}}$$\displaystyle \frac{d}{dx}\text{arccsc}{(x)}=-\frac{1}{|x|\sqrt{x^2-1}}$

Hyperbolic Formulas

• $\frac{d}{dx}\mathrm{sinh}\left(x\right)=\mathrm{cosh}\left(x\right)$$\displaystyle \frac{d}{dx}\sinh{(x)}=\cosh{(x)}$

• $\frac{d}{dx}\mathrm{cosh}\left(x\right)=\mathrm{sinh}\left(x\right)$$\displaystyle \frac{d}{dx}\cosh{(x)}=\sinh{(x)}$

• $\frac{d}{dx}\mathrm{tanh}\left(x\right)={\text{sech}}^{2}\left(x\right)$$\displaystyle \frac{d}{dx}\tanh{(x)}=\text{sech}^2{(x)}$

• $\frac{d}{dx}\mathrm{coth}\left(x\right)=-{\text{csch}}^{2}\left(x\right)$$\displaystyle \frac{d}{dx}\coth{(x)}=-\text{csch}^2{(x)}$

• $\frac{d}{dx}\text{sech}\left(x\right)=-\text{sech}\left(x\right)\mathrm{tanh}\left(x\right)$$\displaystyle \frac{d}{dx}\text{sech}{(x)}=-\text{sech}{(x)}\tanh{(x)}$

• $\frac{d}{dx}\text{csch}\left(x\right)=-\text{csch}\left(x\right)\mathrm{coth}\left(x\right)$$\displaystyle \frac{d}{dx}\text{csch}{(x)}=-\text{csch}{(x)}\coth{(x)}$

Inverse Hyperbolic Formulas

• $\frac{d}{dx}\text{arcsinh}\left(x\right)=\frac{1}{\sqrt{1+{x}^{2}}}$$\displaystyle \frac{d}{dx}\text{arcsinh}{(x)}=\frac{1}{\sqrt{1+x^2}}$

• $\frac{d}{dx}\text{arccosh}\left(x\right)=\frac{1}{\sqrt{{x}^{2}-1}}$$\displaystyle \frac{d}{dx}\text{arccosh}{(x)}=\frac{1}{\sqrt{x^2-1}}$

• $\frac{d}{dx}\text{arctanh}\left(x\right)=\frac{1}{1-{x}^{2}}$$\displaystyle \frac{d}{dx}\text{arctanh}{(x)}=\frac{1}{1-x^2}$

• $\frac{d}{dx}\text{arccoth}\left(x\right)=\frac{1}{1-{x}^{2}}$$\displaystyle \frac{d}{dx}\text{arccoth}{(x)}=\frac{1}{1-x^2}$

• $\frac{d}{dx}\text{arcsech}\left(x\right)=-\frac{1}{x\sqrt{1-{x}^{2}}}$$\displaystyle \frac{d}{dx}\text{arcsech}{(x)}=-\frac{1}{x\sqrt{1-x^2}}$

• $\frac{d}{dx}\text{arccsch}\left(x\right)=-\frac{1}{|x|\sqrt{1+{x}^{2}}}$$\displaystyle \frac{d}{dx}\text{arccsch}{(x)}=-\frac{1}{|x|\sqrt{1+x^2}}$

Antiderivatives

Antidifferentiation Rules

Basic Forms

• $\int {u}^{n}\phantom{\rule{0.167em}{0ex}}du=\frac{{u}^{n+1}}{n+1}+C$$\displaystyle \int{u^n}\,du=\frac{u^{n+1}}{n+1}+C$, $n\ne 1$$n\ne 1$

• $\int {u}^{-1}\phantom{\rule{0.167em}{0ex}}du=\int \frac{1}{u}\phantom{\rule{0.167em}{0ex}}du=\mathrm{ln}|u|+C$$\displaystyle \int{u^{-1}}\,du=\int{\frac{1}{u}}\,du=\ln{|u|}+C$

• $\int {a}^{u}\phantom{\rule{0.167em}{0ex}}du=\frac{{a}^{u}}{\mathrm{ln}\left(a\right)}+C$$\displaystyle \int{a^u}\,du=\frac{a^u}{\ln{(a)}}+C$, $a>0$$a>0$, $a\ne 1$$a\ne 1$

• $\int \mathrm{tan}\left(u\right)\phantom{\rule{0.167em}{0ex}}du=\mathrm{ln}|\mathrm{sec}\left(u\right)|+C$$\displaystyle \int{\tan{(u)}}\,du=\ln{|\sec{(u)}|}+C$

• $\int \mathrm{sec}\left(u\right)\phantom{\rule{0.167em}{0ex}}du=\mathrm{ln}|\mathrm{sec}\left(u\right)+\mathrm{tan}\left(u\right)|+C$$\displaystyle \int{\sec{(u)}}\,du=\ln{|\sec{(u)}+\tan{(u)}|}+C$

• $\int \mathrm{cot}\left(u\right)\phantom{\rule{0.167em}{0ex}}du=\mathrm{ln}|\mathrm{sin}\left(u\right)|+C$$\displaystyle \int{\cot{(u)}}\,du=\ln{|\sin{(u)}|}+C$

• $\int \mathrm{csc}\left(u\right)\phantom{\rule{0.167em}{0ex}}du=\mathrm{ln}|\mathrm{csc}\left(u\right)-\mathrm{cot}\left(u\right)|+C$$\displaystyle \int{\csc{(u)}}\,du=\ln{|\csc{(u)}-\cot{(u)}|}+C$

• $\int \frac{1}{\sqrt{{a}^{2}-{u}^{2}}}\phantom{\rule{0.167em}{0ex}}du={\mathrm{sin}}^{-1}\left(\frac{u}{a}\right)+C$$\displaystyle \int{\frac{1}{\sqrt{a^2-u^2}}}\,du=\sin^{-1}{\left(\frac{u}{a}\right)}+C$

• $\int \frac{1}{{a}^{2}+{u}^{2}}\phantom{\rule{0.167em}{0ex}}du=\frac{1}{a}{\mathrm{tan}}^{-1}\left(\frac{u}{a}\right)+C$$\displaystyle \int{\frac{1}{a^2+u^2}}\,du=\frac{1}{a}\tan^{-1}{\left(\frac{u}{a}\right)}+C$

Integration Strategy

This is a suggested strategy to use while attempting to evaluating an integral. This can be used for indefinite and definite integrals.

Fundamental Theorem of Calculus - Part I

Use known formulas from your knowledge of derivatives to find antiderivatives.

Example 01: $\int \mathrm{sec}\left(x\right)\mathrm{tan}\left(x\right)\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int{\sec{(x)}\tan{(x)}\,dx}$

Since we know $\frac{d}{dx}\mathrm{sec}\left(x\right)=\mathrm{sec}\left(x\right)\mathrm{tan}\left(x\right)$$\dfrac{d}{dx}\sec{(x)}=\sec{(x)}\tan{(x)}$, then

$⇒\int \frac{d}{dx}\mathrm{sec}\left(x\right)\phantom{\rule{0.167em}{0ex}}dx=\int \mathrm{sec}\left(x\right)\mathrm{tan}\left(x\right)\phantom{\rule{0.167em}{0ex}}dx$$\Rightarrow\int{\dfrac{d}{dx}\sec{(x)}\,dx}=\int{\sec{(x)}\tan{(x)}\,dx}$, and

$\mathrm{sec}\left(x\right)+C=\int \mathrm{sec}\left(x\right)\mathrm{tan}\left(x\right)\phantom{\rule{0.167em}{0ex}}dx$$\sec{(x)}+C=\int{\sec{(x)}\tan{(x)}\,dx}$

Thus, $\int \mathrm{sec}\left(x\right)\mathrm{tan}\left(x\right)\phantom{\rule{0.167em}{0ex}}dx=\mathrm{sec}\left(x\right)+C$$\displaystyle \int{\sec{(x)}\tan{(x)}\,dx}=\sec{(x)}+C$

Example 02: $\int {e}^{3x}\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int{e^{3x}\,dx}$

Since we know $\frac{d}{dx}{e}^{3x}=3{e}^{3x}$$\dfrac{d}{dx}e^{3x}=3e^{3x}$, then

$⇒\int \frac{d}{dx}{e}^{3x}\phantom{\rule{0.167em}{0ex}}dx=\int 3{e}^{3x}\phantom{\rule{0.167em}{0ex}}dx$$\Rightarrow\int{\dfrac{d}{dx}e^{3x}\,dx}=\int{3e^{3x}\,dx}$, which is equivalent to

$⇒\int \frac{d}{dx}{e}^{3x}\phantom{\rule{0.167em}{0ex}}dx=3\int {e}^{3x}\phantom{\rule{0.167em}{0ex}}dx$$\Rightarrow\int{\frac{d}{dx}e^{3x}\,dx}=3\int{e^{3x}\,dx}$, and

$⇒\frac{1}{3}\int \frac{d}{dx}{e}^{3x}\phantom{\rule{0.167em}{0ex}}dx=\int {e}^{3x}\phantom{\rule{0.167em}{0ex}}dx$$\Rightarrow\dfrac{1}{3}\int{\frac{d}{dx}e^{3x}\,dx}=\int{e^{3x}\,dx}$

Thus, $\int {e}^{3x}\phantom{\rule{0.167em}{0ex}}dx=\frac{1}{3}{e}^{3x}+C$$\displaystyle \int{e^{3x}\,dx}=\frac{1}{3}e^{3x}+C$

Rewrite the Integrand

Use algebra or trigonometry to rewrite the integrand.

Example 03: $\int \frac{x-1}{{x}^{2}-1}\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int{\frac{x-1}{x^2-1}\,dx}$

$\int \frac{x-1}{{x}^{2}-1}\phantom{\rule{0.167em}{0ex}}dx=\int \frac{x-1}{\left(x-1\right)\left(x+1\right)}\phantom{\rule{0.167em}{0ex}}dx$$\int{\dfrac{x-1}{x^2-1}\,dx}=\int{\dfrac{x-1}{(x-1)(x+1)}\,dx}$

$⇒\int \frac{1}{x+1}\phantom{\rule{0.167em}{0ex}}dx$$\Rightarrow\int{\dfrac{1}{x+1}\,dx}$ $⇒\mathrm{ln}\left(x+1\right)+C$$\Rightarrow\ln{(x+1)}+C$

Example 04: $\int \frac{{x}^{2}}{{x}^{2}-1}\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int{\frac{x^2}{x^2-1}\,dx}$

$\int \frac{{x}^{2}}{{x}^{2}-1}\phantom{\rule{0.167em}{0ex}}dxd=\int \frac{{x}^{2}+0}{{x}^{2}-1}\phantom{\rule{0.167em}{0ex}}dx$$\int{\dfrac{x^2}{x^2-1}\,dx}d=\int{\dfrac{x^2{\color{gsured}+0}}{x^2-1}\,dx}$

$⇒\int \frac{{x}^{2}-1+1}{{x}^{2}-1}\phantom{\rule{0.167em}{0ex}}dx$$\Rightarrow\int{\dfrac{x^2{\color{gsured}-1+1}}{x^2-1}\,dx}$

$⇒\int \left(\frac{{x}^{2}-1}{{x}^{2}-1}+\frac{1}{{x}^{2}-1}\right)\phantom{\rule{0.167em}{0ex}}dx$$\Rightarrow\int{\left(\dfrac{x^2-1}{x^2-1}+\dfrac{1}{x^2-1}\right)\,dx}$

$⇒\int \left(1+\frac{1}{{x}^{2}-1}\right)\phantom{\rule{0.167em}{0ex}}dx$$\Rightarrow\int{\left(1+\dfrac{1}{x^2-1}\right)\,dx}$

$⇒\int \left(1-\frac{1}{1-{x}^{2}}\right)\phantom{\rule{0.167em}{0ex}}dx$$\Rightarrow\int{\left(1{\color{gsured}{-}}\dfrac{1}{{\color{gsured}1-x^2}}\right)\,dx}$

$⇒x-\text{arctanh}\left(x\right)+C$$\Rightarrow x-\text{arctanh}(x)+C$

Example 05: $\int \frac{\mathrm{tan}\left(x\right)}{\mathrm{sin}\left(x\right)}\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int{\frac{\tan{(x)}}{\sin{(x)}}\,dx}$

$\int \frac{\mathrm{tan}\left(x\right)}{\mathrm{sin}\left(x\right)}\phantom{\rule{0.167em}{0ex}}dx=\int \frac{\mathrm{sin}\left(x\right)}{\mathrm{cos}\left(x\right)}\frac{1}{\mathrm{sin}\left(x\right)}\phantom{\rule{0.167em}{0ex}}dx$$\int{\dfrac{\tan{(x)}}{\sin{(x)}}\,dx}=\int{\dfrac{\sin{(x)}}{\cos{(x)}}\dfrac{1}{\sin{(x)}}\,dx}$

$⇒\int \frac{1}{\mathrm{cos}\left(x\right)}\phantom{\rule{0.167em}{0ex}}dx$$\Rightarrow\int{\dfrac{1}{\cos{(x)}}\,dx}$

$⇒\int \mathrm{sec}\left(x\right)\phantom{\rule{0.167em}{0ex}}dx$$\Rightarrow\int{\sec{(x)}\,dx}$

$⇒\mathrm{ln}|\mathrm{sec}\left(x\right)+\mathrm{tan}\left(x\right)|+C$$\Rightarrow\ln{\left|\sec{(x)}+\tan{(x)}\right|}+C$

Example 06: $\int {\mathrm{sin}}^{2}\left(x\right)\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int{\sin^2{(x)}\,dx}$

$\int {\mathrm{sin}}^{2}\left(x\right)\phantom{\rule{0.167em}{0ex}}dx=\int \frac{1}{2}\left(1-\mathrm{cos}\left(2x\right)\right)\phantom{\rule{0.167em}{0ex}}dx$$\int{\sin^2{(x)}\,dx}=\int{\dfrac{1}{2}\left(1-\cos{(2x)}\right)\,dx}$

$⇒\frac{1}{2}\left(x-\frac{1}{2}\mathrm{sin}\left(2x\right)\right)+C$$\Rightarrow\dfrac{1}{2}\left(x-\dfrac{1}{2}\sin{(2x)}\right)+C$

Practice 01

Evaluate the following integrals:

1. $\int \left(3{x}^{2}+x-5\right){e}^{-x}\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int{(3x^2+x-5)e^{-x}\,dx}$

2. $\int \frac{1}{1-{x}^{2}}\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int{\frac{1}{1-x^2}\,dx}$

3. $\int \frac{x}{1-{x}^{2}}\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int{\frac{x}{1-x^2}\,dx}$

4. $\int \frac{{x}^{2}}{1-{x}^{2}}\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int{\frac{x^2}{1-x^2}\,dx}$

5. $\int \frac{{x}^{3}}{1-{x}^{2}}\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int{\frac{x^3}{1-x^2}\,dx}$

6. $\int \frac{{x}^{2}}{1+{x}^{2}}\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int{\frac{x^2}{1+x^2}\,dx}$

7. $\int \frac{{x}^{3}}{1+{x}^{3}}\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int{\frac{x^3}{1+x^3}\,dx}$

8. $\int \frac{1+{x}^{2}}{1+x}\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int{\frac{1+x^2}{1+x}\,dx}$

9. $\int \frac{{\mathrm{sec}}^{2}\left(x\right)}{1+\mathrm{tan}\left(x\right)}\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int{\frac{\sec^2{(x)}}{1+\tan{(x)}}\,dx}$

10. $\int \frac{1-\mathrm{sin}\left(x\right)}{\mathrm{cos}\left(x\right)}\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int{\frac{1-\sin{(x)}}{\cos{(x)}}\,dx}$

11. $\int \frac{x}{{x}^{2}-3x+2}\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int{\frac{x}{x^2-3x+2}\,dx}$

12. $\int \frac{{x}^{2}-3x+2}{x}\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int{\frac{x^2-3x+2}{x}\,dx}$

13. $\int {\mathrm{sin}}^{-1}\left(x\right)\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int{\sin^{-1}{(x)}\,dx}$

14. $\int {\mathrm{sin}}^{3}\left(x\right){\mathrm{cos}}^{5}\left(x\right)\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int{\sin^3{(x)}\cos^5{(x)}\,dx}$

15. $\int {\mathrm{sin}}^{3}\left(x\right){\mathrm{cos}}^{2}\left(x\right)\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int{\sin^3{(x)}\cos^2{(x)}\,dx}$

16. $\int {\mathrm{tan}}^{3}\left(x\right){\mathrm{sec}}^{3}\left(x\right)\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int{\tan^3{(x)}\sec^3{(x)}\,dx}$

17. $\int {\mathrm{tan}}^{4}\left(x\right){\mathrm{sec}}^{4}\left(x\right)\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int{\tan^4{(x)}\sec^4{(x)}\,dx}$

18. $\int {\mathrm{cos}}^{4}\left(x\right)\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int{\cos^4{(x)}\,dx}$

19. $\int \frac{1}{x\sqrt{\mathrm{ln}\left(x\right)}}\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int{\frac{1}{x\sqrt{\ln{(x)}}}\,dx}$

20. $\int x\mathrm{ln}\left(x\right)\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int{x\ln{(x)}\,dx}$

21. $\int {e}^{\sqrt{x}}\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int{e^{\sqrt{x}}\,dx}$

22. $\int \sqrt{{x}^{2}-4x}\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int{\sqrt{x^2-4x}\,dx}$

23. $\int \frac{1}{{x}^{2}\sqrt{{x}^{2}+1}}\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int{\frac{1}{x^2\sqrt{x^2+1}}\,dx}$

24. $\int {e}^{3x}\mathrm{cos}\left(5x\right)\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int{e^{3x}\cos{(5x)}\,dx}$

25. $\int \frac{1}{{e}^{-x}+{e}^{x}}\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int{\frac{1}{e^{-x}+e^x}\,dx}$

26. $\int \frac{x+1}{{x}^{2}+4}\phantom{\rule{0.167em}{0ex}}dx$$\displaystyle \int{\frac{x+1}{x^2+4}\,dx}$