Find the indefinite integral and check the results

An Smith

An Smith

Answered question

2022-05-16

Find the indefinite integral and check the results by differentiation.

dxx7

 

x4+3x2-5x3dx

Answer & Explanation

An Smith

An Smith

Beginner2022-05-16Added 1 answers

Evaluate the following:

.25(4x3 - 8x +7)dxo-π(5cos θ dθ)

 

alenahelenash

alenahelenash

Expert2022-06-05Added 556 answers

1) 1x7dx

Apply basic rules of exponents.

Move x7 out of the denominator by raising it to the -1 power.

(x7)-1dx

Multiply the exponents in (x7)-1.

x-7dx

By the Power Rule, the integral of x-7 with respect to x is -16x-6.

-16x-6+C

Simplify the answer.

Rewrite -16x-6+C as -161x6+C.

-161x6+C

Simplify.

16x6+C

2) x4+3x2-5x3dx

Move x3 out of the denominator by raising it to the -1 power.

(x4+3x2-5)(x3)-1dx

Multiply the exponents in (x3)-1.

Apply the power rule and multiply exponents, (am)n=amn.

(x4+3x2-5)x3-1dx

Multiply 3 by -1.

(x4+3x2-5)x-3dx

Expand (x4+3x2-5)x-3.

x+3x-1-5x-3dx

Split the single integral into multiple integrals.

xdx+3x-1dx+-5x-3dx

By the Power Rule, the integral of x with respect to x is 12x2.

12x2+C+3x-1dx+-5x-3dx

Since 3 is constant with respect to x, move 3 out of the integral.

12x2+C+3x-1dx+-5x-3dx

The integral of x-1 with respect to x is ln(|x|).

12x2+C+3(ln(|x|)+C)+-5x-3dx

Since -5 is constant with respect to x, move -5 out of the integral.

12x2+C+3(ln(|x|)+C)-5x-3dx

By the Power Rule, the integral of x-3 with respect to x is -12x-2.

12x2+C+3(ln(|x|)+C)-5(-12x-2+C)

Simplify.

12x2+C+3(ln(|x|)+C)-5(-12x2+C)

Simplify.

12x2+3ln(|x|)-5(-12x2)+C

Simplify.

12x2+3ln(|x|)+52x2+C

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