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Thread: A few integration problems

  1. #1
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    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.

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

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

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

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

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

    2.$\displaystyle \tan(x)-x+c$

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

    Number 4 I can't solve, can someone help? Thanks
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  2. #2
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    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.

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

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

    $\displaystyle
    \int \sin^3(5x)\cos(5x)dx = K \sin^4(5x) + c
    $

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

    $\displaystyle
    \int \sin^3(5x)\cos(5x)dx = 1/20 \sin^4(5x) + c
    $

    RonL
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  3. #3
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    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.

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

    rewrite your problem to:

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

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

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

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

    EB
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  4. #4
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    Hello, nirva!

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

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

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

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



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

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



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

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

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

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



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

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

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

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

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

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

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  5. #5
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    Quote Originally Posted by nirva View Post

    $\displaystyle 3. \int \frac{\sec^2(x)}{3+\tan(x)}dx$
    Let,
    $\displaystyle u=3+\tan x$
    Then,
    $\displaystyle u'=\sec^2 x$
    Thus, by substitution theorem,
    $\displaystyle \int \frac{u'}{u}du=\int \frac{1}{u}du$
    Which is,
    $\displaystyle \ln |u|+C=\ln |3+\tan x|+C$
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