Find the integrals of the functions in Exercises: ^4x - Vidyadip

Question

Find the integrals of the functions in Exercises:
$\sin^4\text{x}$

✓

Answer

$\sin^4\text{x}=\sin^2\text{x}\sin^2\text{x}$
$=\bigg(\frac{1-\cos2\text{x}}{2}\bigg)\bigg(\frac{1-\cos2\text{x}}{2}\bigg)$
$=\frac{1}{4}(1-\cos2\text{x})^2$
$=\frac{1}{4}\Big[1+\cos^22\text{x}-2\cos2\text{x}\Big]$
$=\frac{1}{4}\Bigg[1+\bigg(\frac{1+\cos4\text{x}}{2}\bigg)-2\cos2\text{x}\Bigg]$
$=\frac{1}{4}\Bigg[1+\frac{1}{2}+\frac{1}{2}\cos4\text{x}-2\cos2\text{x}\Bigg]$
$=\frac{1}{4}\Bigg[\frac{3}{2}+\frac{1}{2}\cos4\text{x}-2\cos2\text{x}\Bigg]$
$\therefore\int\sin^4\text{x}\text{ dx}=\frac{1}{4}\int\bigg[\frac{3}{2}+\frac{1}{2}\cos4\text{x}-2\cos2\text{x}\bigg]\text{ dx}$
$=\frac{1}{4}\bigg[\frac{3}{2}\text{x}+\frac{1}{2}\bigg(\frac{\sin4\text{x}}{4}\bigg)-\frac{2\sin2\text{x}}{2}\bigg]+\text{C}$
$=\frac{1}{8}\bigg[3\text{x}+\frac{\sin4\text{x}}{4}-2\sin2\text{x}\bigg]+\text{C}$
$=\frac{3\text{x}}{8}-\frac{1}{4}\sin2\text{x}+\frac{1}{32}\sin4\text{x}+\text{C}$

Need a full question paper?

Generate a complete, print-ready paper with questions like this in minutes — across 16+ boards, with answer keys.

Start Generating Free

Explore more

More "4 Marks" from INTEGRALS→INTEGRALS — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Let d₁, d₂, d₃ be three mutually exclusive diseases. Let S be the set of observable symptoms of these diseases. A doctor has the following information from a random sample of 5000 patients: 1800 had disease d₁, 2100 has disease d₂, and others had disease d₃. 1500 patients with disease d₁, 1200 patients with disease d₂, and 900 patients with disease d₃ showed the symptom. Which of the diseases is the patient most likely to have?

Let $\text{f}:[-1,\infty)\rightarrow[-1,\infty)$ be given by f(x) = (x + 1)² - 1, $\text{x}\geq-1.$ Show that f is invertible. Also, find the set S = {x : f(x) = f^-1(x)}.

Show that the line $\frac{\text{x}}{\text{a}}+\frac{\text{y}}{\text{b}}=1,$ touches the curve $\text{y}=\text{b}\cdot\text{e}^{\frac{-\text{x}}{\text{a}}}$ a e at the point where the curve intersects the axis of y.

Find the equation of the plane which contains the line of intersection of the planes x + 2y + 3z – 4 = 0 and 2x + y – z + 5 = 0 and whose x-intercept is twice its z-intercept.
Hence write the vector equation of a plane passing through the point (2, 3, –1) and parallel to the plane obtained above.

A hospital dietician wishes to find the cheapest combination of two foods, A and B, that contains at least 0.5 milligram of thiamin and at least 600 calories. Each unit of A contains 0.12 milligram of thiamin and 100 calories, while each unit of B contains 0.10 milligram of thiamin and 150 calories. If each food costs 10 paise per unit, how many units of each should be combined at a minimum cost?

Solve the following differential equation
$\frac{\text{dy}}{\text{dx}}=\text{x}^5\tan^{-1}(\text{x}^3)$

Find the equation of the line passing through the points (1, -1, 1) and perpendicular to the lines joining the points (4, 3, 2), (1, -1, 0) and (1, 2, -1) (2, 1, 1).

Evaluate the following integrals:
$\int(3\text{x}+1)\sqrt{4-3\text{x}-2\text{x}^2}\text{dx}$

Find the equation of the curve through the point (1, 0) if the slope of the tangent to the curve at any point (x, y) is $\frac{\text{y}-1}{\text{x}^2+\text{x}}.$

Express the matrix $\text{A}=\begin{bmatrix}4&2&-1 \\3 & 5&7\\1&-2&1 \end{bmatrix}$ as the sum of a symmetric and a skew-symmetric matrix.