Integration by Substitution (USub)

Use the indicated substitution to evaluate the integral.

 $\int2x\cos(x^2)\ dx\ ,\ \ \ \ u=x^2$  

•  $\sin\left(x^2\right)+C$  

•  $2\sin\left(x^2\right)+C$  

•  $\frac{1}{2}\sin\left(x^2\right)+C$  

•  $4\cos\left(x^2\right)+C$  

Use the indicated substitution to evaluate the integral.

 $\int x\left(x^2+5\right)^7dx\ ,\ \ \ \ u\ =\ x^2+5$  

•  $\frac{\left(2x+5\right)^8}{8}+c$  

•  $\frac{\left(x^2+5\right)^8}{16}+c$  

•  $\frac{\left(x^2+5\right)^6}{6}+c$  

•  $\frac{\left(2x+5\right)^8}{16}+c$  

Use the indicated substitution to evaluate the integral.

 $\int\frac{2x}{\sqrt{3+x^2}}dx\ ,\ \ \ u=3+x^2$  

•  $\frac{1}{2\sqrt{3+x^2}}+C$  

•  $\sqrt{3+x^2}+C$  

•  $2\sqrt{3+x^2}+C$  

•  $\frac{2}{\sqrt{3+x^3}}+C$  

Use the indicated substitution to evaluate the integral.

 $\int-3\cos x\sin\left(\sin x\right)\ dx\ ,\ \ \ u=\sin x$  

•  $3\sec\left(\sin x\right)+C$  

•  $3\sin\left(\sin x\right)+C$  

•  $3\cot\left(\sin x\right)+C$  

•  $3\cos\left(\sin x\right)+C$  

Use substitution to evaluate the integral $\int\frac{2x}{\left(x^2+3\right)^3}\ dx$  

•  $-\frac{1}{2\left(x^2+3\right)^2}+C$  

•  $-\frac{3}{2\left(x^2+3\right)^2}+C$  

•  $-\frac{5}{2\left(x^2+3\right)^2}+C$  

•  $-\frac{1}{\left(x^2+3\right)^4}+C$  

Use substitution to evaluate the integral $\int\left(3x^3+5\right)^5\cdot27x^2dx$ 

•  $\frac{1}{2}\left(3x^3+5\right)^6+C$ 

•  $\frac{2}{5}\left(3x^3+5\right)^5+C$ 

•  $\frac{5}{6}\left(3x^3+5\right)^6+C$ 

•  $\frac{3}{5}\left(3x^3+5\right)^5+C$ 

Use substitution to evaluate the integral $\int4\sec^24x\ \sqrt{\tan4x}\ dx$

• $2\left(\tan4x\right)^{\frac{3}{2}}+C$

• $\frac{15}{4}\cdot\left(\tan4x\right)^{\frac{4}{3}}+C$

• $\frac{4}{3}\cdot\left(\tan4x\right)^{\frac{3}{2}}+C$

• $\frac{2}{3}\cdot\left(\tan4x\right)^{\frac{3}{2}}+C$

Use substitution to evaluate the integral

 $\int\frac{\left(\ln^6x\right)}{x}\ dx$  

•  $7\left(\ln x\right)^7+C$  

•  $\frac{\ln^6x}{x^2}+C$  

•  $\frac{1}{7}\left(\ln x\right)^7+C$  

•  $\frac{6}{7}\left(\ln x\right)^7+C$  

Use substitution to evaluate the integral

 $\int\frac{6\cos t}{\left(2+\sin t\right)^2}dt$  

•  $\frac{6}{2+\sin t}+C$  

•  $\frac{3}{2+\sin t}+C$  

•  $-\frac{6}{2+\sin t}+C$  

•  $-\frac{3}{2+\sin t}+C$  

Use substitution to evaluate the integral

 $\int\ \frac{dx}{x\ln x}$  

•  $\ln\left(\ln x\right)+C$  

•  $\left(\ln x\right)^2+C$  

•  $\frac{1}{x^2}+C$  

•  $\frac{\ln x}{x^2}+C$