Math
Definite Integrals Solutions
Step 1
Submitted
Observe the given integral. Define a function g(x) = (9 - 8x2) and obtain its derivative.
g'(x) = -16 6 -16 x.
Step 2
Now, define f(x) = x. From the given integral, observe the term under the radical sign and rewrite it as
f(g(x)) = f(9 - 8x2)
=
9 – 8x²
9- 8r2
Step 3
Observe that the integrand is of the form f(g(x))g'(x).
Use the Power Rule for Integration,
Jx
xn dx =
x + 1
and Theorem 4.13. The theorem states that,
n + 1
g(x))g'(x) dx = F(g(x)) + C, where F is the antiderivative of f.
Now rewrite the integral.
1/3
| 19 - 8x
9 - 8x?(-16x) dx = (9 - 8x2
1/3
'(-16x) dx
Step 4
Use the Power Rule for Integration to find the antiderivative in terms of u = g(x). Note that we use C for the
constant of integration.
3
3
Juve
1/3 du
344/3
u
+C
4
Step 5
Substituting u = 9 - 8x2 gives
4/3
4/3
(9 - 8
- 8x22
(9 - 8x2)1/3(-16x) dx =
+ C
4
3
(9–82) 19-8-2)***
+ c
4
Step 6
To check the result, differentiate the antiderivative obtained in the previous step by using the Sum Rule of
Differentiation.
(– 3x2)**+c)-(( – 8x?
Bx2)^2)+
(C)
d
8x2))
+O
4
X
Now use the Chain Rule and simplify.
1/3
.:( - )"+c) - 2 0 - x
1/3
(-16x)
1/3
(9-8
22
1/3
(-16)
Step 7
Thus, differentiating the antiderivative gives
the original integrand.
Therefore, the value of indefinite integral
| 19–8x21-162) dx
is
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