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Questions and Answers
What is the result of integrating the function $f(x) = (x^{2} + x)(3x + 1)$?
What substitution is used to simplify the integral $ rac{1}{2} rac{u^{ rac{3}{2}}}{ rac{3}{2}} + C$?
When applying the substitution method, what does $du = g'(x) dx$ represent?
What is the final result of the integration $ rac{1}{7} ext{sin}(u) + C$ when using the substitution $u = 7 heta + 3$?
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What is the correct form of the substitution rule when applying it to integrals?
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Which of the following is the correct substitution for the integral $ ext{sin}(u)du$ when $u = x^3$?
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What constant is included in the result of all indefinite integrals?
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In the integration $ rac{1}{2} ext{int}(2x + 1)^{ rac{1}{2}} dx$, what does the factor $ rac{1}{2}$ correspond to?
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What is the result of the integral $ rac{1}{3} imes rac{2}{5} (2x + 1)^{ rac{5}{2}} - rac{1}{3} imes rac{2}{3} (2x + 1)^{ rac{3}{2}} + C$?
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Which substitution is correctly applied to solve the integral $ rac{1}{2} imes rac{1}{2} imes (1 - rac{1}{4} sin 2x) + C$?
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What is the first step in evaluating the integral $ rac{1}{2} imes rac{2}{5}(2x + 1)^{ rac{5}{2}} - rac{1}{4}(2x + 1)^{ rac{3}{2}} + C$?
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What does the solution $ rac{1}{4} imes (u^{ rac{5}{2}} - u^{ rac{3}{2}}) + C$ represent in context?
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When integrating $ rac{2z}{ ext{sqrt}(z^2 + 1)} dz$, which substitution is used?
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What is the final form of the solution to the integral $ rac{1}{2} imes (1 - cos 2x) dx$?
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What is an appropriate interpretation of $ rac{1}{2} imes rac{1}{2} imes (1 + cos 2x) dx$ in the context of integrals?
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What is a necessary step before integrating the function $sin^2 x$?
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What is the result of the integration $\int 2x (x + 5)^{4} dx$?
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For the integral $\int (3x + 2)(3x^{2} + 4x)^{4} dx$, what does the substitution $u = 3x^{2} + 4x$ imply for $du$?
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What is the final expression after integrating $\int sin(3x) dx$ with the substitution $u = 3x$?
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What does the integral $\int sec(2t) tan(2t) dt$ evaluate to with the substitution $u = 2t$?
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What is the result of the integral $\int \frac{9x^{2}}{\sqrt{1 - x^{3}}} dx$ using the substitution $u = 1 - x^{3}$?
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What is the integral of $\int \sqrt{3 - 2S} dS$ after using the substitution $u = 3 - 2S$?
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What is the expression for the integral $\int \theta \sqrt[4]{1 - \theta^2} d\theta$ using the substitution $u = 1 - \theta^{2}$?
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In the integration process of $\int (3x + 2)(3x^{2} + 4x)^{4} dx$, after substitution, what is the multiplier necessary before the integral of $u^{4}$?
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Study Notes
Indefinite Integrals and Substitution Method
- The indefinite integral of a function $f(x)$, denoted by $\int f(x) dx$, is a family of functions whose derivative is $f(x)$.
- The general form of an indefinite integral is $F(x)+C$, where $F'(x) = f(x)$ and C is an arbitrary constant of integration.
- The substitution method is a technique used to solve indefinite integrals of the form $\int f(g(x)).g'(x)dx$.
- The method involves substituting $u = g(x)$ and $du = g'(x) dx$, transforming the integral into $\int f(u) du$.
Integration by Substitution
- The chain rule in differentiation states that $\frac{d}{dx}(F(g(x))) = F'(g(x)).g'(x)$.
- The substitution method essentially reverses the chain rule, allowing us to integrate functions involving composite functions.
Steps for Substitution Method
- Step 1: Identify a suitable substitution $u = g(x)$.
- Step 2: Find the derivative of $u$ with respect to $x$ and solve for $dx$.
- Step 3: Substitute $u$ and $dx$ in the integral, transforming it into an integral involving only $u$.
- Step 4: Integrate the transformed integral with respect to $u$.
- Step 5: Replace $u$ with the original expression $g(x)$.
Examples of Substitution Method
-
Example 1: $\int (x^{2} + x)(3x + 1)dx$
- Substituting $u = x^{2} + x$ and $du = (3x + 1) dx$ simplifies the integral to $\int u du$.
- The result is $\frac{1}{6}(x^{2} + x)^{6} + C$.
-
Example 2: $\int \sqrt{2x + 1} dx$
- Substitute $u = 2x + 1$ and $du = 2 dx$.
- The resulting integral $\frac{1}{2} \int u^{\frac{1}{2}} du$ evaluates to $(\frac{1}{3}(2x + 1)^{\frac{3}{2}} + C$.
-
Example 3: $\int sec^{2}(5x + 1). 5dx$
- Use the substitution $u = 5x + 1$ and $du = 5 dx$.
- Integrate $\int sec^{2}udu$ to obtain $tanu + C$.
- Finally, replace $u$ with $5x + 1$ to get $tan(5x + 1) + C$.
-
Example 4: $\int cos(7\theta + 3) d\theta $
- Substitute $u = 7\theta + 3$ and $du = 7d\theta$.
- This leads to $\frac{1}{7} \int cos u du$, which evaluates to $\frac{1}{7} sin (7\theta + 3) + C$.
-
Example 5: $\int x^{2} cos x^{3} dx$
- Substitute $u = x^{3}$ and $du = 3x^{2} dx$.
- The integral becomes $\frac{1}{3} \int cos u du$, which evaluates to $\frac{1}{3} sin x^{3} + C$.
-
Example 6: $\int x \sqrt{2x + 1} dx$
- Substitute $u = 2x + 1$, $x = \frac{u-1}{2}$, and $du = 2 dx$.
- Integration leads to $\frac{1}{10}(2x + 1)^{\frac{5}{2}} - \frac{1}{6}(2x + 1)^{\frac{3}{2}} + C$.
-
Example 7:
- a) $\int sin^{2}x dx = \frac{1}{2}x - \frac{1}{4} sin 2x + C$
- b) $\int cos^{2}x dx = \frac{1}{2}x + \frac{1}{4} sin 2x + C$
- c) $\int (1 - 2sinx) sin^{2}xdx = \frac{-1}{8} cos(4x) + C$
-
Example 8: $\int \frac{2z}{\sqrt{z^{2} + 1}} dz$
- Substitute $u = z^{2} + 1$ and $du = 2z dz$.
- The integral becomes $\int u^{\frac{-1}{2}} du$, which evaluates to $2 \sqrt{(z^{2} + 1)} + C$.
Exercises
- 1/245: $\int 2(2x + 4)^{5} dx = \frac{(2x + 4)^{6}}{6} + C$
- 3/245: $\int 2x (x + 5)^{4} dx = \frac{(x + 5)^{5}}{5} + C$
- 5/245: $\int (3x + 2)(3x^{2} + 4x)^{4} dx = \frac{1}{10} (3x^{2} + 4x)^{5} + C$
- 7/245: $\int sin 3x dx = \frac{-1}{3} cos 3x + C$
- 9/245: $\int sec(2t) tan(2t) dt = \frac{1}{2} sec 2t + C$
- 11/245: $\int \frac{9x^{2}}{\sqrt{1 - x^{3}}} dx = -6\sqrt{1 - x^{3}} +C$
- 17/245: $\int \sqrt{3 - 2S} dS = \frac{-1}{3}(3-2S)^{\frac{3}{2}} + C$
- 19/245: $\int \theta \sqrt[4]{1 - \theta^{2}} d\theta = \frac{-1}{2} \int (1 - \theta^{2})^{\frac{1}{4}}.
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Description
This quiz covers the concepts of indefinite integrals and the substitution method used in calculus. It explains the definition of an indefinite integral, the general form, and how to apply the substitution method. Test your understanding of these fundamental calculus techniques.
