Definite Integrals

Two truth and one lie. Which of the following statement is a lie? 

•  $\int_a^b\ f\left(x\right)\ dx$  has a definite value. 

•  $\int_{ }^{ }f\left(x\right)\ dx$  has infinite solutions because of the arbitrary constant  $c$ .  

•  $\int_a^b\ f\left(x\right)\ dx=\ F\left(a\right)-F\left(b\right).\$  

Given that  $\frac{d}{dx}\left[\left(2x+5\right)\sqrt{4x-5}\right]=\frac{12x}{\sqrt{4x-5}},\$  find

 $\int_{ }^{ }\frac{6x}{\sqrt{4x-5}}dx.\$  

•  $\frac{1}{4}\left[\left(2x+5\right)\sqrt{4x-5}\right]+c$  

•  $2\left[\left(2x+5\right)\sqrt{4x-5}\right]+c\$  

•  $\frac{1}{2}\left[\left(2x+5\right)\sqrt{4x-5}\right]+c$  

•  $4\left[\left(2x+5\right)\sqrt{4x-5}\right]+c$  

Find $\int^{ }\left(\frac{15}{2x+3}-\frac{1}{x+1}\right)\ dx\ .\$  

•  $30\ln\left(2x+3\right)-\ln\left(x+1\right)+c$  

•  $\frac{15}{2}\ln\left(2x+3\right)-\ln\left(x+1\right)+c\$  

•  $5\ln\left(2x+3\right)-\ln\left(x+1\right)+c$  

•  $15\ln\left(2x+3\right)-\ln\left(x+1\right)+c$  

Evaluate $\int_1^2\ \left(3-2x^2+\frac{4}{x^3}\right)\ dx,$  leaving your answer in the exact form. 

Evaluate $\int_1^3\ \frac{4}{\left(3x+2\right)^2}\ \ dx,$  leaving your answer in the exact form. 

Evaluate $\int_{\frac{\pi}{4}}^{\frac{\pi}{2}}\ \left(\sin2x-\cos x\right)\ \ dx,$  leaving your answer in three significant figures. 

Evaluate $\int_1^3\ \left(5e^x+e^{-3x}\right)\ dx,\$  leaving your answer in three significant figures. 

Evaluate $\int_0^1\ \frac{1}{3x+1}\ dx,\$  leaving your answer in three significant figures. 

 $\int_{ }^{ }4\sin2x\cos2x\ dx.\$  Find

•  $2\cos4x+c$  

•  $2\sin4x+c\$  

•  $-\frac{1}{2}\cos4x+c$  

•  $8\cos4x+c\$  

Given that  $\frac{d}{dx}\left(x^2\ln x\right)=x+2x\ln x.\$  Which of the following statement is not true? 

•  $\int_{ }^{ }\left(x+2x\ln x\right)dx=x^2\ln x+c$  

•  $\int_{ }^{ }xdx+\int_{ }^{ }2x\ln x\ dx=x^2\ln x+c$  

•  $\int_{ }^{ }xdx+2x\int_{ }^{ }\ln x\ dx=x^2\ln x\ +c$  

•  $2\int_{ }^{ }x\ln x\ dx=x^2\ln x-\int x\ dx\ +c\$