Integration by Substitution Practice

Sample questions

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1. Sample 1difficulty 2/5

   $\displaystyle\int \cos(3x)\,dx =$

   • A

     $3\sin(3x) + C$

   • B

     $\sin(3x) + C$

   • C

     $-\sin(3x)/3 + C$

   • D

     $\dfrac{\sin(3x)}{3} + C$

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   Why

   $u = 3x$, $du = 3\,dx$. $\int \cos u\,du/3 = \sin u/3$.

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2. Sample 2difficulty 2/5

   $\displaystyle\int 5(2x+1)^4\,dx =$

   • A

     $\dfrac{5(2x+1)^5}{5} + C$

   • B

     $\dfrac{5(2x+1)^5}{10} + C$

   • C

     $\dfrac{(2x+1)^5}{2} + C$

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   • D

     $\dfrac{(2x+1)^5}{2} + C$

   Why

   $u = 2x+1$, $du = 2\,dx$. $5 \int u^4\,du/2 = 5 u^5/10 = u^5/2$.

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3. Sample 3difficulty 2/5

   $\displaystyle\int \sin^3 x \cos x\,dx =$

   • A

     $\dfrac{\sin^3 x}{3} + C$

   • B

     $-\dfrac{\cos^4 x}{4} + C$

   • C

     $\dfrac{\cos^4 x}{4} + C$

   • D

     $\dfrac{\sin^4 x}{4} + C$

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   Why

   $u = \sin x$. $\int u^3\,du = u^4/4$.

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4. Sample 4difficulty 2/5

   $\displaystyle\int 2x(x^2 + 1)^3\,dx =$

   • A

     $(x^2+1)^4 + C$

   • B

     $\dfrac{x^4}{4} + C$

   • C

     $\dfrac{(x^2+1)^3}{3} + C$

   • D

     $\dfrac{(x^2+1)^4}{4} + C$

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   Why

   $u = x^2 + 1$, $du = 2x\,dx$. $\int u^3\,du = u^4/4$.

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5. Sample 5difficulty 2/5

   $\displaystyle\int \dfrac{1}{x \ln x}\,dx =$

   • A

     $\dfrac{1}{(\ln x)^2} + C$

   • B

     $\ln(\ln|x|) + C$

   • C

     $\ln(\ln x) + C$

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   • D

     $\ln x \cdot x + C$

   Why

   $u = \ln x$, $du = dx/x$. $\int du/u = \ln|u| = \ln(\ln x)$ (assuming $\ln x > 0$).

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