Answer to Question #104834 in Calculus for SHIVAM KUMAR

Question #104834
Integrate the following functions w.r.t x:
(A) 1\(2x+1)^3\2
(B)sin (2x+3)
(C)cosec (4x)
(D)1\√1-9x^2
(E)1\1+4x^2
1
Expert's answer
2020-03-13T10:11:27-0400

A.) 1(2x+1)32dx\int \frac{1}{(2x+1)^{\frac{3}{2}}} \, dx

=(2x+1)32dx= \int (2x+1)^{- \frac{3}{2}} \, dx

Let us assume that:

2x+1=u2dx=dudx=du22x + 1 = u \\ \Rightarrow 2 \, dx = du \\ \Rightarrow dx = \frac{du}{2}

Now:

Substituting u=2x+1u = 2x + 1 into the integration and integrating with respect to dudu , we have:

=12u32du= \frac{1}{2} \int u^{-\frac{3}{2}} \, du

=12[u1212]+C= \frac{1}{2} \left[ \frac{u^{-\frac{1}{2}}}{- \frac{1}{2}} \right] + C \hspace{1 cm} [ xndx=xn+1n+1+C  \because \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \, \, ( Where CC is a constant of integration) ]

=1u12+C=1u+C= - \frac{1}{u^{\frac{1}{2}}} + C \\ = - \frac{1}{\sqrt{u}} + C

Undo substitution and we have:

=12x+1+C= - \frac{1}{\sqrt{2x+1}} + C \hspace{1 cm} [ u=2x+1\because u = 2x + 1 ]

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B.) sin(2x+3)dx\int sin (2x + 3) \, dx

Let us assume that:

u=2x+3du=2dxdx=du2u = 2x + 3 \\ \Rightarrow du = 2 \, dx \\ \Rightarrow dx = \frac{du}{2}

Now:

Substituting u=2x+3u = 2x + 3 into the integration and integrating with respect to dudu , we have:

=12sin(u)du= \frac{1}{2} \int \sin (u) \, du

=12cos(u)+C= - \frac{1}{2} \cos (u) + C \hspace {1 cm} [ sin(x)dx=cos(x)+C  \because \int \sin (x) \, dx = - \cos (x) + C \,\, (Where CC is a constant of integration) ]

Undo substitution and we have:

=cos(2x+3)2+C[u=2x+3]= - \frac{\cos(2x+3)}{2} + C \hspace {1 cm} \left[ \because u = 2x+ 3 \right]

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C.) cosec(4x)dx\int \cosec (4x) \, dx

Let us assume that:

4x=u4dx=dudx=du44x = u \\ \Rightarrow 4 \, dx = du \\ \Rightarrow dx = \frac{du}{4}

Now:

Substituting 4x=u4x = u into the integration and integrating with respect to dudu , we have:

=14cosec(u)du= \frac{1}{4} \int \cosec (u) \, du

=14lncosec(u)+cot(u)+C= - \frac{1}{4} \ln \mid \cosec (u) + \cot (u) \mid + C \hspace{1 cm} [ cosec(x)dx=lncosec(x)+cot(x)+C  \because \int \cosec (x) \, dx = - \ln \mid \cosec (x) + \cot(x) \mid + C \,\, (Where CC is a constant of integration)]

Undo substitution and we have:

=lncosec(4x)+cot(4x)4+C[u=4x]= - \frac{\ln \mid \cosec (4x) + \cot (4x) \mid }{4} + C \hspace{1 cm} \left[ \because u = 4x \right]

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D.) 119x2dx\int \frac{1}{\sqrt{1 - 9x^2}} \, dx

=19(19x2)dx= \int \frac{1}{\sqrt{9(\frac{1}{9} - x^2)}} \, dx

=131(13)2x2dx= \frac{1}{3} \int \frac{1}{\sqrt{(\frac{1}{3})^2 - x^2}} \, dx

=13sin1(x13)+C= \frac{1}{3} \sin^{-1} \left( \frac{x}{\frac{1}{3}} \right) + C \hspace{1 cm} [ 1a2x2dx=sin1(xa)+C  \because \int \frac{1}{\sqrt{a^2 - x^2}} \, dx = \sin^{-1} \left( \frac{x}{a} \right) + C \,\, (Where CC is a constant of integration)]

=13sin1(3x)+C= \frac{1}{3} \sin^{-1} (3x) + C

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E.) 11+4x2dx\int \frac{1}{1 + 4x^2} \, dx

=14114+x2dx= \frac{1}{4} \int \frac{1}{\frac{1}{4} + x^2} \, dx

=141(12)2+x2dx= \frac{1}{4} \int \frac{1}{(\frac{1}{2})^2 + x^2} \, dx

=14[112tan1(x12)]+C= \frac{1}{4} \left[ \frac{1}{\frac{1}{2}} \tan^{-1} \left( \frac{x}{\frac{1}{2}} \right) \right] + C \hspace{1 cm} [ 1x2+a2dx=1atan1(xa)+C  \because \int \frac{1}{x^2 + a^2} \, dx = \frac{1}{a} \tan^{-1} \left( \frac{x}{a} \right) + C \,\, (Where CC is a constant of integration) ]

=tan1(2x)2+C= \frac{\tan^{-1} (2x)}{2} + C


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Comments

Assignment Expert
02.08.20, 19:35

Dear rachael, please use the panel for submitting new questions.

rachael
02.08.20, 14:56

∫_1^3▒〖( 3x^2 +48)dx

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