"Evaluate the definite integral. ∫−π4π4(x3+x4tan⁡x)dx" was answered by our community

Aneeka Hunt

Answered question

2021-11-10

Evaluate the definite integral.

\int_{- \frac{π}{4}}^{\frac{π}{4}} (x^{3} + x^{4} \tan x) d x

pattererX

Skilled2021-11-11Added 95 answers

Step 1
The given integral is

\int_{- \frac{π}{4}}^{\frac{π}{4}} (x^{3} + x^{4} \tan x) d x

Step 2
Consider the function

f (x) = x^{3} + x^{4} \tan x

.
Check whether the function

f (x) = x^{3} + x^{4} \tan x

is odd or even.
substitute -x for x in

f (x) = x^{3} + x^{4} \tan x

f (- x) = {(- x)}^{3} + {(- x)}^{4} \tan (- x)

= - x^{3} + x^{4} (- \tan x)

= - (x^{3} + x^{4} \tan x)

=-f(x)
Hence, the function

f (x) = x^{3} + x^{4} \tan x

is odd.
Step 3
If f(x) is an odd function and continuous at [−a,a], then

\int_{- a}^{a} f (x) d x = 0

.
By the above result, the value of integral

\int_{- \frac{π}{4}}^{\frac{π}{4}} (x^{3} + x^{4} \tan x) d x

is 0.
That is,

\int_{- \frac{π}{4}}^{\frac{π}{4}} (x^{3} + x^{4} \tan x) d x = 0

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