can anyone help me integrate with respect to x?

Question

i need to integrate these with respect to x but do not understand it could you pease explain how to do it?
∫ x² - 6 dx
∫upper limit 5 lower limit 0 2x^5 + 2x^-5 dx

Answer 1

Hey there!

Here's the answer.

∫ x^2 - 6 dx --> Write the problem.
∫ x^2 dx-∫ 6 dx --> Recall the identity ∫ f(x)±g(x) dx=∫ f(x) dx ± ∫ g(x) dx.
∫ x^2 dx-6∫ 1 dx --> Recall the identity ∫ cf(x) dx = c∫ f(x) dx.
x^3/3-6(x)+C --> Find the integral for each part. Remember that ∫ x^n dx = x^n+1/n+1, where n does not equal to -1.
x^3/3-6x+C --> Simplify the above expression.

So the answer to the first question is x^3/3-6x+C.

∫ upper limit 5 lower limit 0 2x^5+2x^-5 dx --> Write the problem.
∫ 2x^5+2x^-5 dx --> Change the definite integral from the above problem into indefinite integral. It would be much easier to solve and less confusing.
∫ 2x^5 dx +∫ 2x^-5 dx --> Recall the identity ∫ f(x)±g(x) dx=∫ f(x) dx ± ∫ g(x) dx.
2∫ x^5+2∫ x^-5 --> Recall the identity ∫ cf(x) dx = c∫ f(x) dx.
2(x^6/6)+2(x^-4/-4) --> Find the integral for each part. Remember that ∫ x^n dx = x^n+1/n+1, where n does not equal to -1.
[2(5^6/6)+2(5^-4/4)]-[2(0^6/6)+2(0^-4/4)] --> Apply the Fundamental Theorem of Calculus.
Infinity --> Notice that 0 is being raised to -4 power. Since the negative exponent of a number is the exponent of a reciprocal, the reciprocal of 0 is infinity.

So the answer to the second question is infinity or undefined.

I hope it helps!

Answer 2

Integral (x^2-6) dx

You integrate x^2 and -6 seperately, because derivatives have the rule of adding derivatives together.

Integral x^2 dx= 1/3x^3

Integral -6 dx = -6x

Add them and remember to add a constant.

1/3x^3-6x+C

Integral (0 to 5) (2x^5 + 2x^-5) dx

Integral(a to b)f(x)dx = f(b)-f(a)

f(5)-f(0) = Integral(0 to 5)(2x^5+2x^-5)dx

f(0)=0

All we have to do is find f(5)

2(5^5)+2(5^-5)

I got around 6,250.

Answer 3

Question 1
I = ∫ x² - 6 dx between lims. of 0 to 5
I = [x³ / 3 - 6x ] lims. 0 to 5
I = (125/3 - 30) - 0
I = 125/3 - 90/3
I = 35/3

Question 2
I = ∫ 2x^5 + 2.x^(- 5).dx
I = 2x^6 / 6 + 2.x^(- 4) / (- 4) + C
I = (1/3).x^6 - (1/2) / x^4 + C

Answer 4

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