### Home > APCALC > Chapter 6 > Lesson 6.5.1 > Problem6-172

6-172.

Evaluate each expression.

1. $\int _ { 0 } ^ { x ^ { 2 } } \operatorname { cos } ( e ^ { t } ) d t$

Fundamental Theorem of Calculus states that the derivative of an INDEFINITE integral gives you the original function.
But, this is a definite integral.

Be sure to multiply by the derivative of the bounds.

$2x\cos\left(e^{x^2}\right)−\left(0\right)\cos\left(e^0\right)=2x\cos\left(e^{x^2}\right)$

1. $\frac { d } { d x } \int _ { 0 } ^ { \pi } \sqrt { \operatorname { sin } ( t ) } d t$

Refer to the hints in part (a). Or, notice that the definite integral equals a constant. What is the derivative of a constant?

1. $\int _ { 1 } ^ { 10 } f ^ { \prime } ( x ) d x$, where $f ( x ) = \operatorname { cos } ( \pi e ^ { x ^ { 2 } - 100 } ) ( x ) \operatorname { log } ( x )$

$\text{Translation: }\int_{1}^{10}\frac{d}{dx}\left(x\log(x)\left [ \text{cos}\left ( \pi e^{x^{2}-100} \right ) \right ]\right)dx$