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Exercise

$\int\left(x-csc\:x\:cot\:x\right)dx$

Step-by-step Solution

1

Expand the integral $\int\left(x-\csc\left(x\right)\cot\left(x\right)\right)dx$ into $2$ integrals using the sum rule for integrals, to then solve each integral separately

$\int xdx+\int-\csc\left(x\right)\cot\left(x\right)dx$
2

The integral $\int xdx$ results in: $\frac{1}{2}x^2$

$\frac{1}{2}x^2$
3

The integral $\int-\csc\left(x\right)\cot\left(x\right)dx$ results in: $\csc\left(x\right)$

$\csc\left(x\right)$
4

Gather the results of all integrals

$\frac{1}{2}x^2+\csc\left(x\right)$
5

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$

$\frac{1}{2}x^2+\csc\left(x\right)+C_0$

Final answer to the exercise

$\frac{1}{2}x^2+\csc\left(x\right)+C_0$

Try other ways to solve this exercise

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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 Substitution

In calculus, integration by substitution, also known as u-substitution, is a method for finding integrals. Using the fundamental theorem of calculus often requires finding an antiderivative.

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