Evaluate the definite integral, if it exists. $\int_0^{-4} \sqrt{1-2 x} d x$

Use Part 2 of the Fundamental Theorem of Calculus(

$$\int_a^b f(x) d x=F(b)-F(a) \text {, where } F \text { is any antiderivative of } f \text {, that is, } F^{\prime}=f \text {. }$$

) to evaluate the integral, or state that it does not exist.

$$\int_1^e \frac{x^2+x+1}{x} d x$$

Evaluate the definite integral, if it exists. $\int_0^1(2 x-1)^{100} d x$

Evaluate the integral, if it exists.

$$\int \frac{\sec \theta \tan \theta}{1+\sec \theta} d \theta$$

Evaluate the integral, if it exists.

$$\int \frac{e^x}{\left(e^x+1\right) \ln \left(e^x+1\right)} d x$$

Use the properties of integrals to verify each inequality without evaluating the integrals.

$$\frac{\pi}{6} \leqslant \int_{\pi / 6}^{\pi / 2} \sin x d x \leqslant \frac{\pi}{3}$$

Use Part 2 of the Fundamental Theorem of Calculus(

$$\int_a^b f(x) d x=F(b)-F(a) \text {, where } F \text { is any antiderivative of } f \text {, that is, } F^{\prime}=f \text {. }$$

) to evaluate the integral, or state that it does not exist.

$$\int_0^{0.5} \frac{d x}{\sqrt{1-x^2}}$$

Evaluate the integral, if it exists.

$$\int \frac{x^3}{1+x^4} d x$$

Use the properties of integrals to verify each inequality without evaluating the integrals.

$$\int_4^6 \frac{1}{x} d x \leqslant \int_4^6 \frac{1}{8-x} d x$$

Use Part 2 of the Fundamental Theorem of Calculus(

$$\int_a^b f(x) d x=F(b)-F(a) \text {, where } F \text { is any antiderivative of } f \text {, that is, } F^{\prime}=f \text {. }$$

) to evaluate the integral, or state that it does not exist.

$$\int_1^{\sqrt{3}} \frac{6}{1+x^2} d x$$

Use Part 2 of the Fundamental Theorem of Calculus(

$$\int_a^b f(x) d x=F(b)-F(a) \text {, where } F \text { is any antiderivative of } f \text {, that is, } F^{\prime}=f \text {. }$$

) to evaluate the integral, or state that it does not exist.

$$\int_{-e^2}^{-e} \frac{3}{x} d x$$

Evaluate the integral, if it exists.

$$\int \frac{x}{\sqrt{1-x^4}} d x$$

Use the properties of integrals to verify each inequality without evaluating the integrals.

$$\int_1^2 \sqrt{5-x} d x \geqslant \int_1^2 \sqrt{x+1} d x$$

Evaluate the integral by making the given substitution.

$$\int \frac{x+3}{\left(x^2+6 x\right)^2} d x, \quad u=x^2+6 x$$