Calculate integrals of functions

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Calculate integrals using different techniques with examples and detailed solutions and explanations. Also more exercises with solutions are presented at the bottom of the page.
In all examples and exercises, \( c \) represent the constant of integration.


Examples and Their Solutions

Example 1
Evaluate the integral \[ \displaystyle\int 6 \; \cos x \; \sin x \; dx \]



Example 2
Calculate the integral \[ \displaystyle\int x \; \sqrt{x+1} dx \]



Example 3
Evaluate the integral \[ \displaystyle \int \cos^2 x \; dx \]



Example 4
Evaluate the integral \[\displaystyle\int x^3 e^{x^4} \; dx\]



Example 5
Calculate the integral \[\displaystyle \int \dfrac{\sin (2x)}{1-\cos^2(x)} \; dx \]



Example 6
Calculate the integral \[\displaystyle \int (x+\sin x)^2 \; dx \]



Example 7
Calculate the integral \[\displaystyle \int \dfrac{\sin x}{\sin^2 x - 2 cos x + 2} \; dx \]



Example 8
Calculate the integral \[\displaystyle \int \dfrac{1}{x+\sqrt{x+2}} \; dx \]



Example 9
Evaluate the integral \[\displaystyle \int \dfrac{1}{\tan x} \; dx \]



Example 10
Evaluate the integral \[\displaystyle \int \dfrac{1}{x^2 + 2x + 1} \; dx \]



Example 11
Evaluate the integral \[\displaystyle \int \dfrac{1}{x^2+x+1} \; dx \]



Example 12
Evaluate the integral \[\displaystyle \int \dfrac{x^4-2x^2+x}{x^2+x+1} \; dx \]



Example 13
Evaluate the integral \[\displaystyle \int \left(x^3-\dfrac{1}{x^2}\right)^4 \; dx \]



Example 14
Evaluate the integral \[\displaystyle \int \tan^2(x) \; dx \]



Example 15
Evaluate the integral \[\displaystyle \int x^4(4x^5 - 2)^{10} \; dx \]



Example 16
Evaluate the integral \[\displaystyle \int x^2 \arcsin(x) \; dx \]



Example 17
Evaluate the integral \[\displaystyle \int \sqrt x \ln x \; dx \]



Example 18
Evaluate the integral \[\displaystyle \int \dfrac{\sqrt{x+1}}{x} \; dx \]



Example 19
Evaluate the integral \[\displaystyle \int \sin\left(\sqrt{x}\right) \; dx \]



Example 20
Evaluate the integral \[\displaystyle \int \dfrac{1}{e^x+e^{-x}} \; dx \]



Example 21
Evaluate the integral \[\displaystyle \int \log_5 x \; dx \]



Example 22
Evaluate the integral \[\displaystyle \int \dfrac{x^2}{\sqrt{16 - x^2}} \; dx \]



Exercises

Use the table of integrals and the properties above to calculate the following integrals. [Note that you may need to use more than one of the above properties and methods for one integral].

  1. \( \displaystyle \int (\sqrt{x} - \dfrac{x^3}{4} + x \; \ln x ) dx \)

  2. \( \displaystyle \int \sqrt{x+1} dx \)

  3. \( \displaystyle\int \sin^2 x dx \)

  4. \( \displaystyle\int x \cos(x^2) dx \)

  5. \( \displaystyle\int x e^{x^2} dx \)







More References and links

  1. integrals and their applications in calculus.