# Thread:

1. ## Integration by trigonometric substitution

I'm new to this type of question and I can only get so far into the question before I don't know what to do - I haven't been taught more but I very much would like to know.

Use the substitution $x$ = $3\sin\theta$ to show that

$\int_{1.5}^{3} \sqrt{9 - x^2} dx$ = $\int_{a}^{b} k \cos^2\theta d\theta$ where the values of a, b and k are to be found.

Hence evaluate $\int_{1.5}^{3} \sqrt{9 - x^2} dx$.

So far I can say that $x$ = $3\sin\theta$ and $dx$ = $3\cos\theta d\theta$

$\sqrt{9 - (3\sin\theta)^2}$ = $\sqrt{9 - 9\sin^2\theta}$

= $\sqrt{9 (1 - \sin^2\theta)}$
= $\sqrt{9\cos^2\theta}$
= $3\cos\theta$

And this is as far as I can get.

Could someone please help me finish the question and understand how it works from this point onwards?
Any help is greatly appreciated

2. $\int_{1.5}^{3} \sqrt{9 - x^2} dx$ = $\int_{a}^{b} k \cos^2\theta d\theta$ where the values of a, b and k are to be found.

As you have done we let: $x$ = $3\sin\theta$ and $dx$ = $3\cos\theta d\theta$

so

$\int_{1.5}^{3} \sqrt{9 - x^2} dx$ = $\int_{1.5}^{3}\sqrt{9 - (3\sin\theta)^2} 3\cos\theta d\theta$

= $\int_{1.5}^{3} 3\cos \theta \cdot 3\cos\theta d\theta$

= $\int_{1.5}^{3} 9 \cos^2\theta d\theta$
so a=1.5 b=3 k=9

now to evaluate this integral you can use the identity: $\cos^2 x = \frac{1+cos(2x)}{2}$
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 25, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": true, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.9600411057472229, "perplexity_flag": "head"}