Evaluate the given definite integral using the fundamental theorem of calculus.

$\int_1^4 \frac{(\sqrt{x}-1)^{3 / 2}}{\sqrt{x}} d x$

Simplify the fractional expression. (Expressions like these arise in calculus.)

$$\frac { \frac { 1 } { ( x + h ) ^ { 2 } } - \frac { 1 } { x ^ { 2 } } } { h }$$

Evaluate the following integrals using the Fundamental Theorem of Calculus. $\int _ { 0 } ^ { 2 } \left( 3 x ^ { 2 } + 2 x \right) d x$

Simplify the fractional expression. (Expressions like these arise in calculus.)

$$\frac{(x+h)^3-7(x+h)-\left(x^3-7 x\right)}{h}$$

Evaluate the given definite integral using the fundamental theorem of calculus.

$$\int_0^1\left(x^4+3 x^3+1\right) d x$$

Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function $g(x) = \int_3^x e^{t^2 - t}~dt$.

Calculate the integral $\int_1^3 \frac{1}{x^2} d x$ using the Fundamental Theorem of Calculus and (if necessary) properties of the definite integral.

Find $\displaystyle \int_{0}^{1}2xe^{x^{2}} \ dx$ two ways:

(i) Numerically.

(ii) Using the Fundamental Theorem of Calculus.

Evaluate the following definite integrals using the Fundamental Theorem of Calculus.

$$\int _ { \sqrt { 2 } } ^ { 2 } \frac { d x } { x \sqrt { x ^ { 2 } - 1 } }$$

Use Part 2 of the Fundamental Theorem of Calculus to find each definite integral. $\int_{1}^{3} \frac{1}{x^{3}} d x$

Find $\displaystyle\int_0^1 2 x e^{x^2} d x$ two ways: (i) Numerically. (ii) Using the Fundamental Theorem of Calculus.

Consider the integral $\int_{0}^{4}3\sqrt{x}\ dx.$

(b) Use the Fundamental Theorem of Calculus to find the exact value of the definite integral.

Simplify the fractional expression. (Expressions like these arise in calculus.)

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

Use the Fundamental Theorem of Integral Calculus to determine the value of the definite integral. $\int_{0}^{1} \frac{1}{(x+1)^{2}} d x$