Evaluate the integrals below.-4 x²+6а.* dx1 VxT/7b. Sa sin(t) dtIt/3C. Jx²Vx3 + 2 dxTT/4d. *5(1 + tan(t)) sec? t dt0, Find the derivative of the function,1dtt2 + 2g(x) =

a.=∫14x2+6xdx=∫14(x2-12+6x-12)dx=∫14(x32+6x-12)dxintegrating=[x32+132+1+6x-12+1-12+1]14…

Answered: Evaluate the integrals below. -4 x²+6…

Trigonometry (MindTap Course List)

10th Edition

ISBN:9781337278461

Author:Ron Larson

Publisher:Ron Larson

ChapterP: Prerequisites

SectionP.6: Analyzing Graphs Of Functions

Problem 6ECP: Find the average rates of change of f(x)=x2+2x (a) from x1=3 to x2=2 and (b) from x1=2 to x2=0.

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Question

Evaluate the integrals below.
-4 x²+6
а.
* dx
1 Vx
T/7
b. Sa sin(t) dt
It/3
C. Jx²Vx3 + 2 dx
TT/4
d. *5(1 + tan(t)) sec? t dt
0,

Find the derivative of the function,
1
dt
t2 + 2
g(x) =

Expert Solution

Step 1

$a . = \int_{1}^{4} \frac{x^{2} + 6}{\sqrt{x}} d x = \int_{1}^{4} (x^{2 - \frac{1}{2}} + 6 x^{\frac{- 1}{2}}) d x = \int_{1}^{4} (x^{\frac{3}{2}} + 6 x^{\frac{- 1}{2}}) d x i n t e g r a t i n g = {[\frac{x^{\frac{3}{2} + 1}}{\frac{3}{2} + 1} + 6 \frac{x^{\frac{- 1}{2} + 1}}{\frac{- 1}{2} + 1}]}_{1}^{4} = {[\frac{2 x^{\frac{5}{2}}}{5} + 12 x^{\frac{1}{2}}]}_{1}^{4} = \frac{2 {(4)}^{\frac{5}{2}}}{5} + 12 {(4)}^{\frac{1}{2}} - \frac{2 {(1)}^{\frac{5}{2}}}{5} - 12 {(1)}^{\frac{1}{2}} = \frac{64}{5} + 24 - \frac{2}{5} - 12 = \frac{62}{5} + 12 = \frac{122}{5} a n s w e r$

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