Evaluate the following integrals. a)\int \left ( 4\sqrt[3]{x}+\frac{4}{\sqrt{x}}-\csc ^{2}...

Question:

Evaluate the following integrals.

a) {eq}\displaystyle \int \left ( 4\sqrt[3]{x}+\frac{4}{\sqrt{x}}-\csc ^{2} x\right )\;dx {/eq}

b) {eq}\displaystyle \int \frac{x+3}{(x^{2}+6x+7)^{3}}\;dx {/eq}

Integration

Following formulas are used to solve the given question:

{eq}\displaystyle\int x^{n}dx=\dfrac{x^{n+1}}{n+1}+c,n\neq-1\\ \displaystyle\int \dfrac{dx}{\sqrt{x}}=2\sqrt {x}+c\\ \displaystyle\int \csc^{2}x=-\cot x+c {/eq}

Answer and Explanation:

Part(a)

{eq}\displaystyle\int \left ( 4\sqrt[3]{x}+\dfrac{4}{\sqrt{x}}-\csc^{2}x \right )dx\\ =4\displaystyle\int \left ( x \right )^{\frac{1}{3}}dx+4\displaystyle\int \dfrac{1}{\sqrt{x}}dx-\displaystyle\int\csc^{2} x \,dx \\ =4\left [ \dfrac{x^{\frac{1}{3}+1}}{\dfrac{1}{3}+1} \right ]+4\times2\sqrt{x}-\left ( -\cot x \right )+c\\ =4\times\dfrac{3}{4}x^{\frac{4}{3}}+8\sqrt{x}+\cot x+c\\ =3x^{\frac{4}{3}}+8\sqrt{x}+\cot x+c {/eq}

Part(b)

{eq}\displaystyle\int \dfrac{x+3}{\left ( x^{2}+6x+7 \right )}dx\\ {/eq}

Let {eq}\,\,x^{2}+6x+7=t {/eq}

Differentiating both side with respect to t.

{eq}\left ( 2x+6 \right )dx=dt\\ 2\left ( x+3 \right )dx=dt\\ \left ( x+3 \right )dx=\dfrac{1}{2}dt\\ \displaystyle\int \dfrac{x+3}{\left ( x^{2}+6x+7 \right )^{3}}dx=\dfrac{1}{2}\displaystyle\int \dfrac{dt}{\left ( t \right )^{3}}=\dfrac{1}{2}\displaystyle\int t^{-3}dt\\ =\dfrac{1}{2}\left [ \dfrac{t^{-3+1}}{-3+1} \right ]+c\\ =\dfrac{1}{2}\times\dfrac{-1}{2}\times t^{-2}+c\\ =\dfrac{-1}{4t^{2}}+c {/eq}

Plugging the value of t

{eq}\displaystyle\int \dfrac{x+3}{\left ( x^{2}+6x+7 \right )^{3}}dx=-\dfrac{1}{4\left ( x^{2}+6x+7 \right )}+c {/eq}


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Integration Problems in Calculus: Solutions & Examples

from AP Calculus AB & BC: Homework Help Resource

Chapter 13 / Lesson 13
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