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Calculus

Integration of Functions

The Indefinite Integral and Basic Formulas of Integration. Table of Integrals.

Page 1
Problems 1-2
Page 2
Problems 3-6

Definition of the Antiderivative and Indefinite Integral

The function $$F\left( x \right)$$ is called an antiderivative of $$f\left( x \right),$$ if
$F’\left( x \right) = f\left( x \right).$ The family of all antiderivatives of a function $$f\left( x \right)$$ is called the indefinite integral of the function $$f\left( x \right)$$ and is denoted by
$\int {f\left( x \right)dx} .$ Thus, if $$F$$ is a particular antiderivative, we may write
${\int {f\left( x \right)dx} }={ F\left( x \right) + C,}$ where $$C$$ is an arbitrary constant.

Properties of the Indefinite Integral

In formulas given below $$f$$ and $$g$$ are functions of the variable $$x,$$ $$F$$ is an antiderivative of $$f,$$ and $$a, k, C$$ are constants.

• $${\int {\left[ {f\left( x \right) + g\left( x \right)} \right]dx} }$$ $$={ \int {f\left( x \right)dx} }+{ \int {g\left( x \right)dx} }$$
• $${\int {kf\left( x \right)dx} }={ k\int {f\left( x \right)dx} }$$
• $${\int {f\left( {ax} \right)dx} }$$ $$={ {\large\frac{1}{a}\normalsize} F\left( {ax} \right) + C}$$
• $${\int {f\left( {ax + b} \right)dx} }$$ $$={ {\large\frac{1}{a}\normalsize} F\left( {ax + b} \right) + C}$$

Table of Integrals

It’s supposed below that $$a,$$ $$p\left( {p \ne 1} \right),$$ $$C$$ are real constants, $$b$$ is the base of the exponential function ($${b \ne 1,b \gt 0}$$).

 $$\int {adx} = ax + C$$ $$\int {xdx} = {\large\frac{{{x^2}}}{2}\normalsize} + C$$ $$\int {{x^2}dx} = {\large\frac{{{x^3}}}{3}\normalsize} + C$$ $$\int {{x^p}dx} = {\large\frac{{{x^{p + 1}}}}{{p + 1}}\normalsize} + C$$ $$\int {\large\frac{{dx}}{x}\normalsize} = \ln \left| x \right| + C$$ $$\int {{e^x}dx} = {e^x} + C$$ $$\int {{b^x}dx} = {\large\frac{{{b^x}}}{{\ln b}}\normalsize} + C$$ $$\int {\sin xdx} = – \cos x + C$$ $$\int {\cos xdx} = \sin x + C$$ $$\int {\tan xdx} = – {\ln \left| {\cos x} \right|} + C$$ $$\int {\cot xdx} = {\ln \left| {\sin x} \right|} + C$$ $$\int {\sec xdx} = {\ln \left| {\tan\left( {\large\frac{x}{2}\normalsize} + {\large\frac{\pi }{4}\normalsize} \right)} \right|} + C$$ $$\int {\csc xdx} = {\ln \left| {\tan\large\frac{x}{2}\normalsize} \right|} + C$$ $$\int {{\sec^2}xdx} = \tan x + C$$ $$\int {{\csc^2}xdx} = -\cot x + C$$ $$\int {\sec x\tan xdx} = \sec x + C$$ $$\int {\csc x\cot xdx} = -\csc x + C$$ $$\int {\large\frac{{dx}}{{1 + {x^2}}}\normalsize} = \arctan x + C$$ $$\int {\large\frac{{dx}}{{{a^2} + {x^2}}}\normalsize} = {\large\frac{1}{a}\normalsize}\arctan {\large\frac{x}{a}\normalsize} + C$$ $$\int {\large\frac{{dx}}{{1 – {x^2}}}\normalsize} = {\large\frac{1}{2}\normalsize}{\ln \left| {{\large\frac{{1 + x}}{{1 – x}}\normalsize}} \right|} + C$$ $$\int {\large\frac{{dx}}{{{a^2} – {x^2}}}\normalsize} = {\large\frac{1}{{2a}}\normalsize}\ln\left| {\large{\frac{{a + x}}{{a – x}}\normalsize}} \right| + C$$ $$\int {\large\frac{{dx}}{{\sqrt {1 – {x^2}} }}\normalsize} = \arcsin x + C$$ $$\int {\large\frac{{dx}}{{\sqrt {{a^2} – {x^2}} }}\normalsize} = \arcsin {\large\frac{x}{a}\normalsize} + C$$ $$\int {\large\frac{{dx}}{{\sqrt {{x^2} \pm {a^2}} }}\normalsize} = {\ln \left| {x + \sqrt {{x^2} \pm {a^2}} } \right|} + C$$ $$\int {\large\frac{{dx}}{{x\sqrt {{x^2} – 1} }}\normalsize} = {\text{arcsec}\left| x \right|} + C$$ $$\int {\sinh xdx} = \cosh x + C$$ $$\int {\cosh xdx} = \sinh x + C$$ $$\int {{\text{sech}^2}xdx} = \tanh x + C$$ $$\int {{\text{csch}^2}xdx} = -\text{coth}\,x + C$$ $$\int {\text{sech}\,x\tanh xdx} = – {\text{sech}\,x} + C$$ $$\int {\text{csch}\,x\coth xdx} = – {\text{csch}\,x} + C$$ $$\int {\tanh xdx} = {\ln \cosh x} + C$$
 $$\int {adx} = ax + C$$ $$\int {xdx} = {\large\frac{{{x^2}}}{2}\normalsize} + C$$ $$\int {{x^2}dx} = {\large\frac{{{x^3}}}{3}\normalsize} + C$$ $$\int {{x^p}dx} = {\large\frac{{{x^{p + 1}}}}{{p + 1}}\normalsize} + C$$ $$\int {\large\frac{{dx}}{x}\normalsize} = {\ln \left| x \right|} + C$$ $$\int {{e^x}dx} = {e^x} + C$$ $$\int {{b^x}dx} = {\large\frac{{{b^x}}}{{\ln b}}\normalsize} + C$$ $$\int {\sin xdx} = – \cos x + C$$ $$\int {\cos xdx} = \sin x + C$$ $$\int {\tan xdx} = – {\ln \left| {\cos x} \right|} + C$$ $$\int {\cot xdx} = {\ln \left| {\sin x} \right|} + C$$ $$\int {\sec xdx} = {\ln \left| {\tan\left( {\large\frac{x}{2}\normalsize} + {\large\frac{\pi }{4}\normalsize} \right)} \right|} + C$$ $$\int {\csc xdx} = {\ln \left| {\tan\large\frac{x}{2}\normalsize} \right|} + C$$ $$\int {{\sec^2}xdx} = \tan x + C$$ $$\int {{\csc^2}xdx} = -\cot x + C$$ $$\int {\sec x\tan xdx} = \sec x + C$$ $$\int {\csc x\cot xdx} = -\csc x + C$$ $$\int {\large\frac{{dx}}{{1 + {x^2}}}\normalsize} = \arctan x + C$$ $$\int {\large\frac{{dx}}{{{a^2} + {x^2}}}\normalsize} = {\large\frac{1}{a}\normalsize}\arctan {\large\frac{x}{a}\normalsize} + C$$ $$\int {\large\frac{{dx}}{{1 – {x^2}}}\normalsize} = {\large\frac{1}{2}\normalsize}\ln \left| {{\large\frac{{1 + x}}{{1 – x}}\normalsize}} \right| + C$$ $$\int {\large\frac{{dx}}{{{a^2} – {x^2}}}\normalsize} = {\large\frac{1}{{2a}}\normalsize}\ln\left| {\large{\frac{{a + x}}{{a – x}}\normalsize}} \right| + C$$ $$\int {\large\frac{{dx}}{{\sqrt {1 – {x^2}} }}\normalsize} = \arcsin x + C$$ $$\int {\large\frac{{dx}}{{\sqrt {{a^2} – {x^2}} }}\normalsize} = \arcsin {\large\frac{x}{a}\normalsize} + C$$ $$\int {\large\frac{{dx}}{{\sqrt {{x^2} \pm {a^2}} }}\normalsize} = {\ln \left| {x + \sqrt {{x^2} \pm {a^2}} } \right|} + C$$ $$\int {\large\frac{{dx}}{{x\sqrt {{x^2} – 1} }}\normalsize} = \text{arcsec}\left| x \right| + C$$ $$\int {\sinh xdx} = \cosh x + C$$ $$\int {\cosh xdx} = \sinh x + C$$ $$\int {{\text{sech}^2}xdx} = \tanh x + C$$ $$\int {{\text{csch}^2}xdx} = -\text{coth}\,x + C$$ $$\int {\text{sech}\,x\tanh xdx} = – \text{sech}\,x + C$$ $$\int {\text{csch}\,x\coth xdx} = – \text{csch}\,x + C$$ $$\int {\tanh xdx} = \ln \cosh x + C$$

Solved Problems

Click on problem description to see solution.

✓Example 1

Calculate $$\int {\left( {\sqrt x + \sqrt[\large 3\normalsize]{x}} \right)dx}.$$

✓Example 2

Calculate the integral $$\int {\left( {\large\frac{3}{{\sqrt[\large 3\normalsize]{x}}}\normalsize + \large\frac{2}{{\sqrt x }}\normalsize} \right)dx} .$$

✓Example 3

Calculate $$\int {\large\frac{{4dx}}{{2 + 3{x^2}}}\normalsize}.$$

✓Example 4

Find the integral $$\int {\large\frac{{\pi dx}}{{\sqrt {\pi – {x^2}} }}\normalsize}.$$

✓Example 5

Calculate the integral $$\int {{{\tan }^2}xdx}.$$

✓Example 6

Find the integral $$\int {\large\frac{{dx}}{{{\sin^2}2x}}\normalsize}$$ without using a substitution.

Example 1.

Calculate $$\int {\left( {\sqrt x + \sqrt[\large 3\normalsize]{x}} \right)dx}.$$

Solution.

${\int {\left( {\sqrt x + \sqrt[\large 3\normalsize]{x}} \right)dx} } = {\int {\sqrt x dx} + \int {\sqrt[\large 3\normalsize]{x}dx} } = {\int {{x^{\large\frac{1}{2}\normalsize}}dx} + \int {{x^{\large\frac{1}{3}\normalsize}}dx} } = {\frac{{{x^{\large\frac{1}{2}\normalsize + 1}}}}{{\frac{1}{2} + 1}} + \frac{{{x^{\large\frac{1}{3}\normalsize + 1}}}}{{\frac{1}{3} + 1}} + C } = {\frac{{2{x^{\large\frac{3}{2}\normalsize}}}}{3} + \frac{{3{x^{\large\frac{4}{3}\normalsize}}}}{4} } = {\frac{{2\sqrt {{x^3}} }}{3} + \frac{{3\sqrt[\large 3\normalsize]{{{x^4}}}}}{4} + C.}$

Example 2.

Calculate the integral $$\int {\left( {\large\frac{3}{{\sqrt[\large 3\normalsize]{x}}}\normalsize + \large\frac{2}{{\sqrt x }}\normalsize} \right)dx} .$$

Solution.

Transforming the integrand and using the formula for integral of the power function, we have
${\int {\left( {\frac{3}{{\sqrt[\large 3\normalsize]{x}}} + \frac{2}{{\sqrt x }}} \right)dx} } = {\int {\frac{{3dx}}{{\sqrt[\large 3\normalsize]{x}}}} + \int {\frac{{2dx}}{{\sqrt x }}} } = {3\int {{x^{ – \large\frac{1}{3}\normalsize}}dx} + 2\int {{x^{ – \large\frac{1}{2}\normalsize}}dx} } = {3 \cdot \frac{{{x^{ – \large\frac{1}{3}\normalsize + 1}}}}{{ – \large\frac{1}{3}\normalsize + 1}} }+{ 2 \cdot \frac{{{x^{ – \large\frac{1}{2}\normalsize + 1}}}}{{ – \frac{1}{2} + 1}} }+{ C } = {\frac{{9{x^{\large\frac{2}{3}\normalsize}}}}{2} + 4{x^{\large\frac{1}{2}\normalsize}} + C } = {\frac{{9\sqrt[\large 3\normalsize]{{{x^2}}}}}{2} + 4\sqrt x + C.}$

Page 1
Problems 1-2
Page 2
Problems 3-6