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f(x)=(6x^2-2x+1)/(4x^3+x)−6−5−4−3−2−10123456−3-2.5−2-1.5−1-0.500.511.522.53xy

Exercise

Step-by-step Solution

Learn how to solve integration by trigonometric substitution problems step by step online. Find the integral int((6x^2-2x+1)/(4x^3+x))dx. Rewrite the expression \frac{6x^2-2x+1}{4x^3+x} inside the integral in factored form. Rewrite the fraction \frac{6x^2-2x+1}{x\left(4x^2+1\right)} in 2 simpler fractions using partial fraction decomposition. Expand the integral \int\left(\frac{1}{x}+\frac{2x-2}{4x^2+1}\right)dx into 2 integrals using the sum rule for integrals, to then solve each integral separately. The integral \int\frac{1}{x}dx results in: \ln\left(x\right).
Find the integral int((6x^2-2x+1)/(4x^3+x))dx

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Final answer to the exercise

ln⁡∣x∣−arctan⁡(2x)+14ln⁡∣4x2+1∣+C0\ln\left|x\right|-\arctan\left(2x\right)+\frac{1}{4}\ln\left|4x^2+1\right|+C_0

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  • Integrate by partial fractions
  • Integrate by substitution
  • Integrate by parts
  • Integrate using tabular integration
  • Integrate by trigonometric substitution
  • Weierstrass Substitution
  • Integrate using trigonometric identities
  • Integrate using basic integrals
  • Product of Binomials with Common Term
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Main Topic: Integration by Trigonometric Substitution

Trigonometric substitution is the substitution of trigonometric functions for other expressions. One may use the trigonometric identities to simplify certain integrals containing radical expressions.

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∫6x2− 2x+14x 3 +x dx
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