WolframAlpha Blogから

「ひとつずつ追う答え」(Step-by-Step solution)はWolfram|Aphlaで数学関連の最も人気のある特長の一つですが、機能的に劇的に拡張されました!新しいインターフェースとともに、ひとつずつ追う答えのすべてを、一度に1段階だけ見るように自分のペースで実行することができます。私たちのプログラムのいくつかは答えを実行する時ヒントを使ってあなたをガイドしようとするでしょう。よくある数学の問題では、解を見つける複数の方法を示すことさえできます。また私たちは、方程式の解、有理算術、三角恒等式の確認という3つの新しい数学コンテンツ領域を紹介することに興奮しています。3つの領域ではひとつずつ追う答えをつかうことができます。Wolfram|Alphaにサインインすると、この新しい特長を一日に3回使うことができます。Wolfram|Alpha Proにアップグレードすると、好きなだけ何度でも使うことができますよ!

Step-by-step solutions, one of the most popular features for mathematics in Wolfram|Alpha, has just received a dramatic expansion in its functionality! With our new interface, you now have the ability to walk through all of our Step-by-step solutions at your own pace, revealing only one step at a time. Some of our programs will offer to guide you with hints when walking through solutions. And for common math problems, we can even show multiple ways to find the solutions. We are also very excited to introduce three new math content areas that now have Step-by-step solutions: solving equations, rational arithmetic, and verifying trigonometric identities. When you’re signed into Wolfram|Alpha, you can use this new feature three times a day. Or, when you upgrade to Wolfram|Alpha Pro, you can use it as many times as you like!

新しい「ひとつずつ追う答え」を積分(最もよくある問い合わせの1つ)について見てみましょう。Wolfram|Alphaに“integrate cos^2(x)”とタイプして結果ページの右上にあるStep-by-step solutionボタンをクリックします。

Let’s look at a new Step-by-step solution for an integral (one of the more popular math queries we receive). We’ll type “integrate cos^2(x)” into Wolfram|Alpha and then click the Step-by-step solution button in the top right of the results page.

問題を一度に1段階実行するためには、Next stepボタンをクリックします。上の画像では既にそうしています。一度に全部見るなら、Show all stepsボタンをクリックします:

To walk through the problem one step at a time, you can click the Next step button, as we have done above. Or if you’d rather see everything at once, click the Show all steps button:

さて、入力(8 * 11) / 3 + 4を見てみましょう。これは私たちのできたてのプログラムからひとつずつ追う答えの特長を見せてくれます。

Now let’s look at the input (8 * 11) / 3 + 4, which features a Step-by-step solution from one of our brand-new programs. In this walkthrough, you will have the option to use hints to help guide you through the problem:

問題を実行する時、ヒントとして次に何が来るかアイデアを与えます。ヒントを使わないなら、右上のHide hintsボタンをクリックします。もちろん、一度にすべての段階を見たいなら、最初の例でやったように"Show all step"をクリックできます。

As you walk through the problem, hints will give you an idea of what comes next. If you’d rather not use the hints, you can click the Hide hints button in the top right. And of course, if you’d like to see all of the steps at once, we can click “Show all steps,” as we did in our first example.


Wolfram|Alpha’s capability to show steps to solve an equation has grown tremendously over the summer! To see this, let’s start by finding the roots of a polynomial:

段階的解のウィンドウの右上隅にはドロップダウンメニューがあり、問題をどう解くか選べます: 因子法を使う、平方根を完成させる、二次公式を使う。3つすべてをやってみて比較しましょう:

The top-right corner of the Step-by-step solutions window has a drop-down menu to let us choose how to solve the problem: use the factor method, complete the square, or use the quadratic formula. Let’s try all three and compare:


Again, we see that we have the option to walk through the steps one at a time (using hints if we’d like) or to show all steps at once.

ヒントや問題を解く複数の方法を提供する以外に、実数上と複素数上で解を解くこともできます!Wolfram|Alphaに (e^x + 2)(x – 1)の根を見つけるよう質問してこれが動くところを見てみましょう。実数上で解く時、Wolfram|Alphaは (e^x + 2)(x – 1)が1つだけ根を持つことを示します; 複素数上ではWolfram|Alphaはこの式の複素数根を見つけます。

In addition to offering hints and multiple methods to solve a problem, we can now solve equations over the real numbers or over the complex numbers! Let’s see this in action by asking Wolfram|Alpha to find the roots of (e^x + 2)(x – 1). When solving over the real numbers, Wolfram|Alpha will show us that (e^x + 2)(x – 1) has only one root; over the complex numbers, Wolfram|Alpha will find the complex roots of this expression.

できたての機能をさらに見るため、Wolfram|Alphaに三角関数の恒等式を検証するよう頼みましょう。これをするには、証明したい恒等式を単にWolfram|Alphaにタイプします。一度に一段階ずつ証明を実行してくれるでしょう。例えば、恒等式 (sin(x) – tan(x))(cos(x) – cot(x)) = (sin(x) – 1)(cos(x) – 1)を試してみましょう:

To see even more of our brand-new functionality, let’s ask Wolfram|Alpha to verify a trigonometric identity. To do this, we simply type the identity we wish to prove into Wolfram|Alpha, and it will walk us through our proof one step at a time. For example, let’s try the identity (sin(x) – tan(x))(cos(x) – cot(x)) = (sin(x) – 1)(cos(x) – 1):


Here are some more examples for you to explore the scope of Step-by-step solutions.


  • 60431 / 89
  • 6876 + 9782 + 816
  • 9 (3+1) + 17 / (6-12)


  • 4x – 6 = 2x+8 の解
  • 実数上で(9^(x + 1)) – (28 (3^(x))) + 3 = 0を解く
  • solve: x^4 + 2x^3 + 5x^2 + 10x + 25 = 0

Polynomial expansion:

  • expanded form of (x + 3)^2
  • expand (x + 1)(x – 1)(x + 2)
  • multiply out (x^2 + x)^4

Partial fraction decomposition:

  • partial fraction decomposition 1/(x^2 + 4x + 3)
  • 1 / (x^3 + 4x^2 + 5x + 2)
  • 1 / (x^4 + 8x^3 + 22x^2 + 24x + 9) partial fraction expansion

Matrix row reduction:

  • {{1,1,5},{1,-1,1}} row reduce
  • reduced row echelon form: {{1, -3, 3, -4}, {2, 3, -1, 15}, {4, -3, -1, 19}}

Proving trigonometric identities:

  • (1 + tan(x))/(1 – tan(x)) = (cos(x) + sin(x))/(cos(x) – sin(x))
  • sin(x)^4 – cos(x)^4 = 1 – 2 cos(x)^2
  • cot(t/2)^2 = (1 + cos(t)) / (1 – cos(t))


  • limit of (x – 3) / (x^2 – 2x – 3) as x approaches 3
  • (e^x – 1 – x) / x^2 as x goes to 0
  • take the limit as x goes to infinity: (1 + 1/x)^x


  • derivative of x^4 + 9x^3 + 7x – 2
  • sin(square root of x) derivative
  • slope of log(x) / (x^2 + 1)


  • integrate sin(x)cos(x)^2
  • integral of sqrt(a^2 – x^2)
  • show the integration: arcsec(sqrt(t))

Ordinary differential equations:

  • y‘(t) – 2y(t) = 3 e^(2t)
  • y”(t) + y(t) = sin(t)
  • t^2 y‘(t) + 2t y(t) = t^4 y(t)^2 + 4

これは新しい「ひとつずつ追う答え」で何ができるか簡単な概要を与えます。Wolfram|Alphaにサインインするとこの新しい特長を一日に3回使うことができます。Wolfram|Alpha Proユーザーは段階的解に制限なしにアクセスできます。

This gives you a brief overview of what you can do with our new Step-by-step solutions. When you’re signed into Wolfram|Alpha, you can use this new feature three times a day. Wolfram|Alpha Pro users receive unlimited access to Step-by-step solutions.


With Wolfram|Alpha’s Step-by-step Solutions feature, you can be guided―at your own pace―through a broad range of math problems, from arithmetic and equation solving all the way through integrals and ordinary differential equations. We look forward to expanding our Step-by-step solutions to more areas―please let us know if there are new solutions that you’d like to see!