Evaluate the following integrals: ^4x ^3x dx - Vidyadip

Question

Evaluate the following integrals:
$\int\sin^4\text{x}\cos^3\text{x}\text{ dx}$

✓

Answer

$\int\sin^4\text{x}\cos^3\text{x}\text{ dx}$
$=\int\sin^4\text{x}\cdot\cos^2\text{x }\cos\text{x}\text{ dx}$
$=\int\sin^4\text{x}\big(1-\sin^2\text{x}\big)\cos\text{x}\text{ dx}$
Let $\sin\text{x}=\text{t}$
$\cos\text{x}\text{ dx}=\text{dt}$
Now, $\int\sin^4\text{x}\big(1-\sin^2\text{x}\big)\cos\text{x}\text{ dx}$
$=\int\text{t}^4(1-\text{t}^2)\text{dt}$
$=\int(\text{t}^4+\text{t}^6)=\text{dt}$
$=\frac{\text{t}^5}{5}-\frac{\text{t}^7}{7}+\text{C}$
$=\frac{\sin^5\text{x}}{5}-\frac{\sin^7\text{x}}{7}+\text{C}$

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 "3 Marks Question" from Indefinite Integrals→Indefinite Integrals — all question types→Maths STD 12 Science question papers→Gujarat Board STD 12 Science question papers→Gujarat Board question paper generator→

Similar questions

Differential equation $\frac{\text{dy}}{\text{dx}}+\text{y}=0,\text{y}(0)=3$Function $\text{y}=\text{e}^\text{-x}+2$

Let N be the set of natural numbers and R be the relation on N × N defined by (a, b) R (c, d) iff ad = bc for all a, b, c, d $\in$ N. Show that R is an equivalence relation.

Sand is pouring from a pipe at the rate of $12\ cm^3/s.$ The falling sand forms a cone on the ground in such a way that the height of the cone is always one-sixth of the radius of the base. How fast is the height of the sand cone increasing when the height is 4cm?

If $\vec{\text{a}},\vec{\text{b}}$ are two vectors such that $\big|\vec{\text{a}}+\vec{\text{b}}\big|=\big|\vec{\text{b}}\big|,$ then prove that $\vec{\text{a}}+2\vec{\text{b}}$ is perpendicular to $\vec{\text{a}}.$

Write a value of $\int\frac{\log\text{x}^{\text{n}}}{\text{x}}\text{ dx}$

Integrate the function in Exercise:
$\frac{\sin\text{x}}{\sin(\text{x}-\text{a)}}$

Find the second order derivatives of the following functions:$\text{x}^3+\tan\text{x}$

Two dice are thrown and it is known that the first die shows a 6. Find the probability that the sum of the numbers showing on two dice is 7.

Using determinants show that the following points are collinear:
(2, 3), (-1, -2) and (5, 8)

Find the equation of the plane passing throught the point (2, 4, 6) and making equal intercepts on the coordinate axes.