Results 1 to 5 of 5

Math Help - A few integration problems

  1. #1
    Newbie
    Joined
    Mar 2006
    Posts
    15

    A few integration problems

    I tried some of these problems and came up with some answers, but not 100% sure about it, please help me check them.

    1. \int \sin^3(5x)\cos(5x)dx

    2. \int \tan^2(x)dx

    3. \int \frac{\sec^2(x)}{3+\tan(x)}dx

    4. \int \sin(x)(\cos(x)+\csc(x))dx

    These are what I got:
    1. -\frac{5}{2}\cos^2(5x)+\frac{15}{4}\cos^4(5x)+c

    2. \tan(x)-x+c

    3. Ln(3+\tan(x))+c

    Number 4 I can't solve, can someone help? Thanks
    Follow Math Help Forum on Facebook and Google+

  2. #2
    Grand Panjandrum
    Joined
    Nov 2005
    From
    someplace
    Posts
    14,972
    Thanks
    4
    Quote Originally Posted by nirva View Post
    I tried some of these problems and came up with some answers, but not 100% sure about it, please help me check them.

    1. \int \sin^3(5x)\cos(5x)dx

    These are what I got:
    1. -\frac{5}{2}\cos^2(5x)+\frac{15}{4}\cos^4(5x)+c
    Observe that 5 \cos(5x) is the derivative of \sin(5x) suggests that the integral is for some K:

    <br />
\int \sin^3(5x)\cos(5x)dx = K \sin^4(5x) + c <br />

    and differentiating the RHS tells us that K=1/20 so:

    <br />
\int \sin^3(5x)\cos(5x)dx = 1/20 \sin^4(5x) + c <br />

    RonL
    Follow Math Help Forum on Facebook and Google+

  3. #3
    Super Member
    earboth's Avatar
    Joined
    Jan 2006
    From
    Germany
    Posts
    5,830
    Thanks
    123
    Quote Originally Posted by nirva View Post
    I tried some of these problems and came up with some answers, but not 100% sure about it, please help me check them.

    ...
    4. \int \sin(x)(\cos(x)+\csc(x))dx
    ...
    Number 4 I can't solve, can someone help? Thanks
    Hello, nirva,

    rewrite your problem to:

     \int \sin(x)(\cos(x)+\csc(x))dx= \int \left( \sin(x)\cdot \cos(x) + \sin(x) \cdot \frac{1}{\sin(x)} \right)dx

    Remember that cos(x) is the derivative of sin(x). Then use substitution:

     \int \left( \sin(x)\cdot \cos(x) + \sin(x) \cdot \frac{1}{\sin(x)} \right)dx= \int \sin(x)\cdot \cos(x) dx+ \int \frac{\sin(x)}{\sin(x)} dx

    (For confirmation only: I've got 1/2*sin^2(x) + x)

    EB
    Follow Math Help Forum on Facebook and Google+

  4. #4
    Super Member

    Joined
    May 2006
    From
    Lexington, MA (USA)
    Posts
    11,806
    Thanks
    697
    Hello, nirva!

    1)\;\int \sin^35x\cos5x\,dx

    Let u = \sin5x\quad\Rightarrow\quad du = 5\cos5x\,dx\quad\Rightarrow\quad dx = \frac{du}{5\cos5x}

    Substitute: . \int u^3\cos5x\left(\frac{du}{5\cos5x}\right) \;=\;\frac{1}{5}\int u^3\,du\;=\;\frac{1}{5}\cdot\frac{u^4}{4} + C

    Back-substitute: . \frac{1}{20}\sin^45x + C



    2. \int \tan^2x\,dx

    We have: . \int\left(\sec^2x - 1\right)\,dx \;=\;\tan x - x + C



    3. \int \frac{\sec^2x}{3+\tan x}dx

    Let u = 3 + \tan x\quad\Rightarrow\quad du = \sec^2x\,dx\quad\Rightarrow\quad dx = \frac{du}{\sec^2x}

    Substitute: . \int\frac{\sec^2x}{u}\left(\frac{du}{\sec^2x}\righ  t) \;= \;\int\frac{du}{u} \;=\;\ln|u| + C

    Back-substitute: . \ln|3 + \tan x| + C



    4. \int \sin x \,\left[\cos x +\csc x \right]\,dx

    Multiply: . \sin x\,\left[\cos x + \csc x\right] \;= \;(\sin x)(\cos x) + (\sin x)(\csc x) \;= \;\sin x \cos x + 1

    We have: . \int\left(\sin x\cos x + 1\right)\,dx \;= \;\int\sin x\cos x\,dx + \int dx

    . . In the first integral, let u = \sin x\quad\Rightarrow\quad du = \cos x\,dx

    . . Substitute: . \int u\,du \;=\;\frac{1}{2}u^2 + c \;= \;\frac{1}{2}\sin^2x + C

    Answer: . \frac{1}{2}\sin^2x + x + C

    Follow Math Help Forum on Facebook and Google+

  5. #5
    Global Moderator

    Joined
    Nov 2005
    From
    New York City
    Posts
    10,616
    Thanks
    9
    Quote Originally Posted by nirva View Post

    3. \int \frac{\sec^2(x)}{3+\tan(x)}dx
    Let,
    u=3+\tan x
    Then,
    u'=\sec^2 x
    Thus, by substitution theorem,
    \int \frac{u'}{u}du=\int \frac{1}{u}du
    Which is,
    \ln |u|+C=\ln |3+\tan x|+C
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Replies: 3
    Last Post: November 3rd 2010, 12:54 AM
  2. Problems with integration word problems
    Posted in the Calculus Forum
    Replies: 5
    Last Post: April 25th 2010, 05:39 PM
  3. integration problems
    Posted in the Calculus Forum
    Replies: 1
    Last Post: April 20th 2009, 04:01 AM
  4. Integration problems
    Posted in the Calculus Forum
    Replies: 2
    Last Post: February 26th 2008, 12:02 AM
  5. Problems with some integration
    Posted in the Calculus Forum
    Replies: 21
    Last Post: July 25th 2006, 03:16 AM

Search Tags


/mathhelpforum @mathhelpforum